Browsers:
Google Chrome
Firefox
The Stock browser on your device.

Music players: (paid)
Poweramp
PlayerPro
N7 Player

Music players: (Free)
Winamp
DoubleTwist
JetAudio

Video:
Dice Player
MX Player (free, ad supported. has a paid ad free version)
Archos (free, ad supported. has a paid ad free version)

Camera Apps:
Camera Zoom FX (paid)
Paper Camera (paid)
Camera 360 ultimate

Image editors:
Photoshop (paid. has diff versions for tabletsm phones)
Picsay pro (free limited version, paid full version)
Pixlr Express

Twitter:
Falcon Pro (paid)
Carbon
Plume (free with ad, paid one without ads)

Launchers:
Apex Launcher (free, requires 4.0+. paid version unlocks more features)
Nova Launcher (free, requires 4.0+. paid version unlocks more features)
Go Launcher
ADW Launcher (paid)

News Reader (including RSS apps)
Flipboard
Google Currents
Feedly

File Explorer:
Solid Explorer (paid, has a ton of features including root support). I like the dual pane view.
ES File explorer (free, and still has a ton of features including root support)
X-plore File Manager (free, and still has a ton of features including root support). Dual pane view

Keyboard:
Swiftkey (paid, diff versions for tablets and phone. buy the tablet version for your Galaxy Tab 2)
Swype (free, can be downloaded from their website only)
Touchpal (free)

Other apps:
Alarm Clock Extreme
Business Calendar (paid. expensive but I got it on sale for 25 cents)
Call Logs Backup & Restore
FB Reader (ebooks)
FTPDroid (ftp client)
Google Authenticator (2 step verification)
Google+ (best social network, its actually quite informative & educational)
Google Keep
Hotmail or Outlook.
Kingsoft Office (great for editing viewing docs of various formats, has pdf support too. free)
Play Books (you can buy books from the Play store)
Play Movies (for buying books from Google Play)
Rescan media (useful when you download new media files, and your media player doesn;t detect it. we no longer have to reboot the device to see the files)
SMS Backup & Restore
Tablet

Wait for sales for the paid versions if you find them too expensive. Same goes for games. Let me know what genre you like