Here's my list. Sorry it's rather long.
Handcent SMS - I use it for texting and love it.
Quick Settings - Quickly change many phone settings from one screen.
Opera Mini - Replacement web browser. They claim it's faster because they cache web pages on their servers for quicker downloads.
CalWidget - I like it for the way it displays my calendar
Google Calendar Sync - I use it to sync my Outlook calendar (for work - Microsoft Exchange) to my Google calendar. When I retire, I will quit using this.
aCar - Record gasoline purchases and vehicle maintenance
handyCalc - powerful calculator
GDocs Notepad - Record notes. I like it because it syncs with my Google Docs account.
Pandora - online music source
Facebook & Twitter - obviously. I use and like the "official" apps for both sites.
Antennas - see where your cell tower is on a Goggle map
Astro File Manager - Manage your files - move, delete, etc.
Color Flashlight - turns your screen bright white plus has many fun effects
Flashlight - turns on the LED flash
Compass - name says it all
GasBuddy - app that goes with gasbuddy.com - find nearby gas prices
PdaNet - tether the phone to your laptop for Internet access. You do not pay an extra fee to your carrier.
Radar Now - displays weather radar for your current location. Knows where you are via GPS.
TipCalc - for calculating tips in many different ways.
WiFi Analyzer - Finds and measures the strength of wifi signals.
Winamp - Android music player from the developers who brought us the famous music player for our computers
All the above are free, although a few have a paid version that doesn't have ads.
These two are paid apps.
PowerAMP $4.99 - Another music player, but this one has a graphical equalizer which makes a HUGE improvement in the sound from your headphones or speakers. You can download a 15-day trial version from the Market.
SwiftKey "On sale" for $1.99 the last time I looked - Keyboard replacement. I love it for its ability to predict the next word. It learns from my previous typing. There is a free version where the predictions expire after 31 days.
I'm sorry this list is so long, but have fun testing out these apps
