The easiest way to start downloading apps onto your Android device is to go to the Google Play Store (usually pre-installed on your device) and to choose from the compatible apps listed there.
You could also go to play.google.com on your computer (if you don't have a Google account you will need to sign up for one, don't worry it's free). The Google play store has thousands of apps which are free or free to try. Apps are split into categories (I used my own for the links below) making it easy to find stuff you need.
The variety of apps is very great, showing us that almost anything is possible with an Android device.
Category: Messaging
Category: Office
- Aldiko Book Reader
- ColorNote Notepad
- Cool Reader
- Dropbox
- ESV Bible
- Google Calendar
- Google Drive
- Skydrive
- 3G Watchdog
- Avast Mobile Security
- Dolphin Browser
- Gmail
- Google+
- Kobo
- Opera Mini Browser
- Skype
- Youtube
- ES File Explorer
- Line Camera
- MX Player
- Photo Art Colour Effects
- Retro Camera
- Smart Voice Recorder
- Winamp
Category: Tools
A special mention to Tasker, which is an amazing app. Sadly, my photo does not have enough RAM or CPU power to use it.
At a later stage I will zoom in on a few apps, at the moment there is not enough time for me to do a writeup of each of them.
Feel free to mention your favourite apps in the comments.