This list is a little dated, There use to be a sticky of good apps on this board, but I am not sure where it has gotten off to. There is also a whole section on Apps in the forum.
Communication
Google Voice - saves voicemail to your GMail account and transcribes them to text
Handcent - SMS replacement, adds features and notifications
Meebo IM- all-in-one IM suite
Missed Call - custom ring and vibrate settings as well as LED color and settings
SMS Popup - replacement for Android SMS notifier (turn stock off to prevent doubles)
Save MMS - allows you to save MMS attachments
Twidroid - Twitter
WiFi Buddy - detailed WiFi information and settings
Music and sound
Album Art Grabber - gets missing album art for your music
Frequency-Generator - sine, square, saw, and triangle wave generator
gStrings - customizable instrument tuner; also works for voice and other sounds
Instant Lyrics - open while playing a song and will open its lyrics
MixZing - music player with advanced category features and album art finder
Scrobble Droid - enables Last.fm scrobbling in the Android music player
Pandora - online radio
Shazam - analyzes a part of a song that's playing through speakers and tells you the name and artist
Slacker Radio - online radio with Twitter integration and song/artist details
Images
PaintNote - MS Paint-like app
Photoshop Mobile - basic editor and online album
PicSay - multitouch image viewer/editor
Location
Car Finder - press once when you park and saves your location on its map, and will show your location as well so you can find your parking spot
Wheres My Droid - lets you turn up the volume and activate ring and GPS to locate your phone; uses an SMS code phrase
Google Places Directory - lists categories of places in your location; searchable
GPS Status - 360 degree compass, horizontal and vertical level, g-force sensor, magnetic field sensor, GPS location in long/lat, shows speed and elevation
Reference
aCar - stores your car information
Barcode Scanner - scans various types of bar codes and Google searches them
ColorNote - text, drawing, and voice note program
ColorDict - comprehensive dictionary with multiple add-ons available
PDFGView - PDF viewer
ShopSavvy - barcode scanner with built-in product lookup
Speed Test - Ookla's speedtest.net application
Unit Converter
Money
Bank of America - online banking and account management
FX Currency - comprehensive currency converter
Google Finance - stocks
PayPal
Security and other
DockRunner - enables multimedia dock mode without needing the dock
Wheres My Droid - see above
G-backup - backs up almost anything you want to Google's servers
MyIP - shows your phone's IP
Text-To-Speech Toy - converts input text to speech
Games
Hyperspace - sort of like Super Monkey Ball; tilt the Droid to navigate a ball around the level within the time limit
Labyrinth - tilt the Droid to roll the ball around holes and into the goal
Quake - full Quake game as far as I know, runs very well with full graphics
