Social
Twidroid
In my opinion the best twitter app on android. Feel free to ty the alternatives though, of which "TwitterRide", "Seesmic" and "Swift" are leading the rest of the pack.
Pro Version
Free (Limited) version
Facebook for Android
Official facebook app, currently the most stable. Checkout Bloo if you fancy a change.
Listen
Googles own podcast app, very good app itself, but those who take their podcasting a bit more seriously should opt for doggcatcher below. Use this instead of doggcatcher if you don't use it enough to warrant paying for a podcast app.
DoggCatcher
Strange icon but a fantastic podcast app, with a hard working developer adding lots of updates. I personally use this for my podcasts. Well worth the money over the free google listen.
Meebo IM
Possibly the best IM app on the android atm. Uses data not SMS. AIM MSN Yahoo Facebook MySpace Google Talk Jabber and ICQ available.
AndChat
Must have if you want to connect to IRC on the go. Best IRC app on the market, and very supportive developer.
System Tools
Force Close Button
Adds a button to the pull-down notifications list that, when selected, forces the current app to close. Use this app A LOT.
AppManager
Many options to manage apps including backup and uninstall.
aTrackDog
Tracks updates to your apps
Startup Auditor
Some apps add themselves to the startup of your phone when you don't want them to, use this to disable those apps.
TasKiller
Use to check what apps are currently running, and kill/auto-kill any tasks. Works like your task manager on a PC would. Please don't over use this as most of the time YOU DON'T NEED TO CLOSE RUNNING APPS!
Full Version
Free (Ad-Supported) Version
Astro
Top notch file manager, probably the best on the android right now.
HideNSeek
Hide media files on your phone, disguised as a flashlight app
Locale
Dynamically manages your phone settings based on conditions, such as Location and Time.
Bar Control
Lets you to put links to apps in the notification bar.
Volume Control
Control your audio settings very easily.
Web
TorrentFu
Scan a barcode, and this will automatically find the torrent for that item, and add it to your torrent client on your home computer! Alternative is Transdroid.
FTPServer
Turns your phone into an FTP server so you can transfer files to and from your phone wirelessly from your PC. NOTE: Use SDReScan once you have transferred files to make your phone rescan your sdcard and notice the new files you added.
AndFTP
The opposite of FTPServer. This is an FTP client that allows you to connect to FTP servers.
Remember there are new apps coming out everyday, so just drop me an email or tweet if you think others would find a different app beneficial.