Most applications available on the Android Market are free. If there is a charge it is usually for a version with extra features and without advertisements.
Is it worth buying these? That is a question only you can answer. Do you want those features? Do you hate the ads?
But consider this. Most paid versions of applications cost around the price of a cup of good coffee. So don't sweat it any more than the decision to go and buy a coffee when you are in town, tired and thirsty.
I'm a great Open Source fan. I also run an Ad Blocker. But I pay for lots of Apps (after testing the free version) even if I don't really need those extra features. The 'Coffee Test' comes into play and also the fact that I want to encourage people to keep developing good applications for Android.
Here are a few applications I recommend that every Android user should have on their device.
I have no relationship with the people who are associated with the applications I recommend below.
This Application requires a Rooted device
Always back up your setting etc. before doing anything out of the ordinary to your Android device. On a Rooted device your Recovery ROM will allow you to backup your ROM. For everything else, at least on a Rooted device, there is really only one App,  Titanium Backup and the associated Pro and Media Sync addons. Use it.
The 'Launcher' is the part of Android that handles the desktop screens and icons, the App Draw etc. The standard version is a little limited and many people use an after-market version. I follow CyantogenMod and use  ADW Launcher , paying for the extras provided by the  EX version (its only the price of a cup of coffee).
The gains in terms of functionality, flexibility and personalisation are wonderful.
Most Android Apps remain in memory once nominally 'closed'. Many will continue to consume CPU (and hence battery) power. This slows down your device and can eat into your battery time. What is needed is a way to automatically 'kill' these still running Apps.
The latest versions of Android are supposed to have this functionality built in but it never seems to work as well as expected so there is still a place for after-market App Killers.
I use two:
There are many applications available that improve on the native battery capacity display. However Battery Snap monitors your usage pattern and gives you an estimate of how long that battery capacity will last you. It also monitors charging and gives you an estimate of time remaining to full charge based on the amps being delivered and your usage. I find it so useful I wouldn't be without it.
There is also an add-on, that I haven't tested, that integrates location data into these estimates, so if you don't use your device often at work this will be taken into account.
Tip
The Battery Snap Widget may stop updating from time to time. If it does, tap on it to enter the Battery Snap application and select one of the options that shows you graphical data. Then back out again. The Widget shout be updating again.
Updated 7 January 2012
Everybody has their own preferences for soft keyboards but obviously people didn't much like the soft keyboard provided with the earlier versions of Android as it was significantly revamped for Android 2.3 (Gingerbread). Most manufacturers provide their own, improved, version.
I find the Gingerbread keyboard is still insufficiently flexible and lacks an important feature, arrow keys. Without arrow keys, editing mistakes in the centre of a string can be really difficult to handle, with ultra-precision tap control required to select the right point. With arrows it becomes a doddle.
So I use  Smart Keyboard PRO, which is very easy to customise to your preferences and has, optional, arrow keys. Don't forget to also add the  English dictionary (or another, more appropriate, one, if you want) so that the auto-correct feature works properly.
This Application requires a Rooted device
 AdFree is an Ad Blocker. It gets rid of those nasty adverts that annoy you in the free versions of software. Keep it up to date and 99% of those go away. Life is now so much less stressful.
All devices should have protection from viruses.  Antivirus Free is the Mobile version of the popular AVG visrus checker available for the PC. This is the fre version, other variants are available.
Added 7 January 2012
Most Android devices have a quantity of internal storage and one or more MicroSD memory cards. The internal storage is frequently quite small relative to that on the MicroSD card.
By default applications are installed to the internal storage. If you have a lot of apps or a small internal memory area this can limit the number of apps you can install. As a result Android has the ability to move applications from the interanal to SD card memory.
Not all applications can run from the SD card, for example Widgets will not function. Plus access to the dialogs required to make the move is quite fiddly. On top of this updates will install back to internal memory.
 Apps 2 SD makes managing this much easier, with a specialist interface and the ability to warn you when you install and app or update that could be moved to the SD card.
Added 7 January 2012
Some of the most used or useful settings on Android are buried deep in the stack of menus.  Quick Settings allows you to access them quickly though a customisable menu.
Added 7 January 2012
Its often useful to change sets of settings en-block. For example at work I want my tablet silent with the WiFi off.  Quick Profiles allows you to set up groups of settings ('profile') and change them with a single selection.
Tip
Make changing the wallpaper part of each profile. This makes it much easier to tell at a glance which profile you are in.
Added 7 January 2012
These are the applications that I find to be 'Best of the Bunch' for certain tasks. I have added a date so you can see how recent my evaluation was.
task: | Twitter Client |
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app: |  UberSocial for Twitter |
comment: | Better than the next best, TweetDeck <https://play.google.com/store/apps/details?id=com.thedeck.android.app>, but still clunky in some areas. |
date: | 5 April 2012 |
task: | Web Browser |
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app: |  Dolphin Browser |
comment: | Several variations available, pick according to device and taste. Many Plugins available. |
date: | 7 January 2012 |
task: | Text to Voice |
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app: |  Classic Text to Speech Engine |
comment: | Allows your phone to speak English, not American. Don't forget to also download the relevant voices. |
date: | 7 January 2012 |
task: | Alarm Clock |
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app: |  Alarm Droid |
comment: | Don't miss the existence of the Text to Voice system, which can be used instead or as well as the conventional tones. Also has a 'Speacking Clock' mode. Can be slightly complex to set up, but that’s the price you pay for flexibility. |
date: | 7 January 2012 |
task: | Digital Clock |
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app: |  Advanced Digital Clock |
comment: | Flexible, clear and without extraneous animations. |
date: | 7 January 2012 |
task: | Time Recording |
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app: |  Time Recording - Timesheet App |
comment: | Also has a 'Pro' version, which is what I use. |
date: | 7 January 2012 |
task: | GPS Suport |
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app: |  GPS Status & Toolbox |
comment: | Check how your GPS is getting on plus a tools to speed up getting the fix. |
date: | 7 January 2012 |
task: | Mapping (1) |
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app: |  RMaps |
comment: | Can use many map sources, including Ordnance Survey. Consider using the Open Street Map variations, which include specialist maps for cyclists and are Open Source - if you finad an error you can correct it yourself. |
date: | 7 January 2012 |
task: | Mapping (2) |
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app: |  BackCountry Navigator PRO GPS |
comment: | Fewer map sources than RMaps, even for the OS, but I get on with the interface slightly better. |
date: | 7 January 2012 |
task: | Satellite Navigation |
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app: |  CoPilot Live UK + Ireland |
comment: | 'Premium' version now available, but I haven't worked out what the differences are yet. |
date: | 7 January 2012 |
task: | Weather Forecast |
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app: |  UK Weather |
comment: | Interface to Met Office data. The 'Favourites' interface is a bit non-obvious but this is much easier than browsing the Met Office website. |
date: | 7 January 2012 |
task: | Shipping Forecast |
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app: |  Marine Weather: UK Edition |
comment: | Access to the latest Inshore and Offshore Shipping Forecasts. |
date: | 7 January 2012 |
task: | File Transfer |
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app: |  Dropbox |
comment: | 'Cloud Storage', this is just one offering amongst many but is widely used and supported. Use with Dropsync to automatically transfer and synchronise files between devices and PC's without the annoyances of mounting your device as storage. |
date: | 7 January 2012 |
task: | Directory Synchronisation |
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app: |  Dropsync |
comment: | I use this with Dropbox to do things like automatically transfer photos from phone to PC. |
date: | 7 January 2012 |
task: | Adobe Acrobat Document Reading |
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app: |  ezPDF Reader |
comment: | Much better for reading .pdf format books than the Adobe offering. This is the free 'Lite' version, you probably want the 'Pro'. |
date: | 7 January 2012 |
task: | RSS Feed Reader |
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app: |  gReader |
comment: | I have only just begun to use this application, it does seem, however, to be the best currently available. |
date: | 7 January 2012 |
task: | FTP Client |
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app: |  AndFTP |
comment: | Looks clunky but it is very effective, better than the alternatives I have tried. |
date: | 20 March 2012 |
Added 7 Jan 2012 Changed 5 Apr 2012
Caution
Always backup you settings (with Titanium Backup) before changing them.
If you are getting lots of 'This application is not compatible with your device' messages on the market then it may be a consequence of the Dots Per Inch (DPI) setting for your device. Many applications are programmed, or declared to the Market, in such a way that they only work with one of the standard Android DPI settings, 120, 160, 240 or 320, or a subset of these. If your device has a DPI setting not allowed by the App, say it is to low or non-standard, then you will get the 'Not compatible' message.
So, check your DPI setting. Or, more practically, reset it to one of the standard settings. The easiest way is to download the  DPI Changer Utility Choose the setting according to taste, or closest to the original setting for your device, but always one that allows you to use the applications you want.
Be aware that changing the DPI will change the 'zoom' level you see on your screen and may make the display less (or more) clear.
Most of the time you will use your Android device on WiFi networks where an IP address is assigned without your intervention, via a system known as DHCP. There may be, however, circumstances where you want to set a specific IP address (and if, at this point, you don't know what I'm talking about you probably don't)
Android allows you to set an IP for the phone, although this will apply to all networks. The menus are accessible via Settings ‣ Wireless & networks ‣ WiFi settings ‣ Menu ‣ Advanced
If you want to change the IP settings on a per network basis without having to dive deep into the menus and remembering those arcane network details then you will have to use a WiFi Manager that changes the IP Settings for you. I use  WiFi Manager Premium, but I am far from convinced that it is the best application for this use around.
By default Android devices announce themselves to the network using an internally generated Host Name based on their MAC Address. This can be a pain if you are trying to keep up with what is happening on your network.
However on some devices (and I'm not sure whether its down to my CyanogenMod ROM or the simple fact that this uses a later version of Android) you can set the Host Name via an entry under Settings ‣ Applications ‣ Development.
Danger
Rooting and Custom ROMs
Rooting your Android device or installing a Custom ROM can leave your device in an unusable state. It will almost certianly invalidate your warranty.
Do not do this unless you are confident of what you are doing and can bear to take the consequences. With some devices the procedures are far from straightforward, check and check again, review what you will be doing all the way through before you do it and search the net for any little wrinkles before you do it.
I accept no responsibility if you try this and it goes wrong.
That said many people do this and the results can be very rewarding. Good Luck!
'Rooting' is a process that allows you to get full access to all the features of you phone. As standard Android limits your access to various aspects of the Operating System in order to prevent you breaking it and/or subverting things like Digital Rights Management software.
As Android is Linux based a 'Rooted' phone gives the owner access as the 'root' user, hence the name.
Depending on the phone the process can be easy or difficult. Getting things wrong can result in an unstable or damaged phone.
There is more information at  Wikipedia
Added 7 Jan 2012
The term 'ROM' is a bit of a misnomer, but lets skate over that...
Custom ROMs are versions of the Android Operating System developed by the user community. They typically provide one or more of the following:
It is necessary to first 'Root' a phone before installing a Custom ROM
Custom ROMs used to be deeply frowned upon by manufacturers, who went to great lengths to block Rooting as a result. But increasingly manufacturers are changing their attitude, to the extent that some are starting to provide support for this activity.
Many ROMs are available, with the number available for a device typically depending on its popularity with the tech-savy community. Most are available from the  xdadevelopers on the devices area in the  forums, look for the 'Android Development' subsection for the device.
Two groups specialise in producing similar ROMs for a large number of devices,  CyanogenMod and  Modaco. These are always good quality ROMs, if sometimes not quite 'up-to-the-bleeding-edge-'. Both have a set of 'Officially Supported' devices and a loosely affiliated group of users who take the 'Official' release and make it available for less popular devices. I try to use CyanogenMod ROMs whenever they are available, indeed the availability of a CyanogenMod ROM is a major factor in my buying decisions.
Added 7 Jan 2012
I have recently (December 2011) encountered two low costs Android Tablets and can recommend both.
See the warning regarding Rooting and Custom ROMs
This is a lot of tablet for not much money. Currently (December 2011) selling for around £175.00, it's basic but functional, suitable for the non power user (like my mother, who received one for Christmas 2011). There is not a lot more to say than that, it does the job!
It ships with 2.2 but that can be changed to 2.2.2 with a custom ROM, a variety of which are available from the wonderful  xda . A port of the  CyanogenMod 7.1 ROM is being worked on - when it arrives pounce on it, CyanogenMod ROMs are very much the 'quality' end of the custom ROM market and will add  significant features as well as upgrading you to Android 2.3.x (Gingerbread).
A new version of the ViewPad, the 7K, is due out soon, featuring Android 2.3 (Gingerbread) as standard. When it does I expect it will be priced above the ViewPad 7 and for the older model to largely disappear from the market. Get yours whilst it is still available.
This is an 'own brand' offering from  Currys Digital available for around £200.0 in December 2011. The upside is that it has a 10.1 inch screen and is cheap, the downside is that it lacks the ability to connect to the Mobile network and has some idiosyncratic design features that essentially require you to install a custom ROM in order to make it properly usable. However as a cheap, large format, tablet for use around a home with WiFi it is great, provided you are willing to install that custom ROM.
The idiosyncratic features are:
Probably because of these idiosyncrasies the convenient  Rom Manager from ClockworkMod will not work with this tablet. You will just have to do things the old way...
There are a variety of ROMs out there for the Vega but I used  CyanogenMod 7.1 for Advent Vega which is officially still 'Unstable' but works well apart from a 'Force Close' on the DSP App at startup. I am loving it.
If, like me, you have a home network you really ought to have a Firewall. Heck, even if you have only one machine, you ought to have a Firewall.
Things like the Windows Firewall are all well and good when you have one machine to deal with but if you want to do anything much more complex then a separate device is useful. Plus having a Firewall running on the thing it is protecting is inherently weak.
If you want something better you should consider using  IPCop which takes a standard PC (perhaps your old, retired one) and turns it into a Firewall/Router. Its what I have used for many years.
The basic list of extra features you get over running a Firewall on your main machine are:
Passphrases for WiFi devices can be a major pain. Particularly if they include punctuation characters. One way I make them more manageable is to partition the phrase into sections of a few characters, say 4, and ensure that they are all of one class, Lower Case, Upper Case, Number or Punctuation. It makes entering them so much more manageable, especially if your software designer decided that you didn't need to actually _see_ what you had entered as that 64 character random string.
So a 24 character string might be 'abcd ABCD 1234 !"£$% efgh EFGH' (obviously not a real Passphase). Keep the pattern you use as secret as the Passphrase, it is, in effect part of it.
Obviously this is not as random as a completely randomised string so, unless you already use a maximum length string, allow some extra characters.
If you want some extra security there is no reason why your partitioning should be regular or the blocks all of one type. For example you could have 'abcdef ABC 1234 !"£$%^ ghijkl' or 'aA1! bB2" cC3£ dD4$ eE4% fF5%'
There are many random Passphrase generators. Most of them, however suffer from one or both of two big security holes:
There is one site that partially gets round this iis the one from  Really Effective. The phrases are transmitted in clear but several are given and you can pick, randomly or otherwise, which one you use.
This is in no way perfect, it would be much better to use a local generator, but it's a start to solving the problem.
You could also achieve the same effect by using another generator several times.
The generator is also highly customisable.
If you are setting up or modifying a WiFi network you should be aware that  Edimax Wireless Range Extenders (and, quite probably, those from some other manufacturers) mangle the MAC Address of devices connected to them. So if your firewall implements MAC Address filtering you may have problems.
One work round is, of course, use the Range Extenders own internal MAC Address filtering and disable the feature on your firewall.