Here is my "must have" apps list. Many are free, but some do cost money. Apps that cost money are marked with a $
4Shared - You can keep items in the 4Shared cloud for easy download from your PC or device. Also compatible with 4Sync.
B&N Nook - the standard reader for Barnes & Noble ebooks. Can also read pdf files.
Beautiful Widgets ($) - A lot of customization features and widgets. make your phone your own.
Dolphin Browser - A great fully featured browser for free. Several add-ons are available to enhance your experience as well.
FuelLog - Track all of your expenses for your car.
Gas Buddy - look up the best prices for gas near you.
Jota Text Editor - a free fully featured text editor.
Kingsoft Office - Free word processor, powerpoint and spreadsheet manager. Great compatibility with PC documents
MapQuest - a great alternative to Google Maps
Maxthon Browser - another great browser with full screen browsing and other tools
PowerAMP (Trial and $) - an awesome music center with several tools, like an equalizer, sound leveling, presets and more
NetFlix - a must if you have a subscription
Redbox - see what your local Redbox has in stock. You can even reserve and set up notifications in the app.
WinAmp - a free app that can sync with your PC's version
Advanced Task Manager - So you can kill apps that are misbehaving quickly
App2SD - See and transfer any apps that can be stored on your SD card.
Battery Indicator - puts your battery's percentage left in your notification bar
BetterBatteryStat ($) - a tool that lets you monitor what is using your battery while in standby. Can really help maximize battery life.
JuiceDefender - Can auto-manage various features on your phone to maximize battery life.
Passdroid - Encrypted local storage for all your passwords.
Quick App Install - Good for installing apps that are from outside sources.
Rom Toolbox (Free and $) - A must for root users. So many great tools all in one.
Root Explorer ($) - A file explorer for rooted phones.
SD Maid (Free and $) - Several tools that can help you manage your internal (root needed) and external (SD card) memory.
Titanium Backup (Free and $) - A backup tool for rooted users. Paid version can also freeze bloatware and integrate updates into system apps.
Angry Birds (there are now 5 games in the series) - launch birds at pigs that have stolen their eggs. Great physics engines in all games. Premium version available for Space and in app content for all. ($)
Bad Piggies - play from the pig's point of view to steal the eggs
Bard's Tale ($) - a PS2 and PC classic, ported to Android. Right now it is half off ($2.99)
ChuChu Rocket ($) - A mindbending strategy game, help the ChuChu make it to their rockets while avoiding the KapuKapu's.
Game Dev Story (Trial and $) - Manage your own software company to the top of the industry.
Great Little War Game ($) - Lead your troops into battle.
Great Big War Game ($) - See above.
Plants vs Zombies ($) - Defend your home from hoards of zombies and your crazy neighbor...
and last but not least:
Phandroid - you are registered here aren't you?