These are my recommended apps, let's hear yours in the comments!
Dolphin Browser is the fastest browser I've come across, and has a decent UI. Plug-ins are also accepted here, so you can even run Flash in Dolphin if you want.
Glovebox is an app that gives you quick access to a bunch of apps (up to 8 in the free version) from swiping the edge of the screen.
Light Flow is amazing - you can customize the LED notification light on your phone so that each notification type illuminates the LED a different color.
SwiftKey ($4) is the best keyboard app I've used. After trying out a ton of keyboards, I always come back to SwiftKey.
Where's My Droid lets you locate your phone. Through SMS messages, you can
remotely make your phone ring, flash the LED, and vibrate, or send a
text back with GPS coordinates.
Action Launcher Pro ($4.29) is the
way I like to setup my home screen. It puts your app tray on the left
(accessible via a swipe), and also has 'Covers' (folders hidden under
apps), ans 'Shutters' (swipe up from an app to launch it's widget.
Nexus Triangles ($1.01) is everything you want a Live Wallpaper to be. Fast, smooth, doesn't hog battery, but most of all it looks amazing and has the options to tweak it to your liking.
NFS Most Wanted ($4.99) is my go-to game on my phone. It runs smooth, with great controls. A must have for any racing junkie.
SG: DeadZone is the only first person shooter app I have installed.
It runs pretty smooth and is a giant online chaos game. It's also free.
Super Hexagon ($1.07) has been distracting me from more important things. It is a
simple concept, retro graphics, amazing soundtrack, and is the most
difficult game I have downloaded.
Podkicker ($2.99) is the only app I use to listen to podcasts. It's free, simple, and have a very usable UI.
Imgur for Android easily lets you upload images to imgur, so it's easier
than ever to upload an image to Reddit from your phone/tablet.
QuickPic has the best UI of any gallery app I've used and the only thing it lacks is the ability to show PhotoSpheres.
RealCalc improves upon the stock calculator so much, you can't afford not to download this free app.
Keep is made by Google, and is an extremely simple notes app which auto syncs with Google Drive.
Reddit is Fun has all the bells and whistles you could want from a mobile Reddit browser.
Reddit Now is a Reddit client that uses the Holo theme, although it does miss some functionality.
Self Serve - I'm with Virgin Mobile (Canada) and this is their app which informs you of how much of your monthly plan you've used (talk, text, or data). See if your provider has a similar app, it could save you money.
PVSTAR+ is a youTube player which still runs in the background, and allows you to watch videos not meant for mobile devices.
MX Player can play any file format you can think of, with a nice UI.