I use and like:
 The official facebook app
 Twidroyd  (twitter)
 k-9 mail (e-mail application that works better than my built-in app)
 Ringdroid (ringtone / alarm creator)
 ConnectBot (ssh client)
 Advanced Task Killer (for use with Android version 2.1 and earlier)
 Watchdog (for use with Android version 2.2)
 Astrid (task list)
 Barcode Scanner (reads barcodes and does clever things with them)
 ShopSavvy (also reads barcodes, then does searching for you)
 Audible (native Audible client for audible books, if applicable)
 Shazam (same as on the iPhone)
 Yelp (native Yelp client)
 Alarm Klock (nice alarm app)
 Pandora
 Kindle (kindle client, if you use amazon kindle books)
 3G Watchdog (if you want/need to track your data use)
 Dropbox (for accessing your dropbox on your phone)
 Vibrate During Meeting (makes the phone vibrate or go silent when you are marked as "busy" on your calendar)
 Our Groceries (keep a shopping list between multiple phones)
 AK Notepad
 handyCalc (a better calculator)
 UltraChron (a stopwatch)
 Vignette (a better camera app than on the Droid X)
 
  Google apps:
     Latitude (if you want selected others to know where your phone is)
     Calendar (nice integration with Google Calendar, if you use that)
     Navigation (turn by turn GPS directions)
     Maps (google maps client)
     Places (points on interest through google)
     Youtube (youtube client)
     Voice
    Sky (fucking amazing "the future is here" sky map.  See something in
the sky that you want to identify? Run the app and hold the phone up to
the sky, and the app tells you what you are seeing!)
  I'm starting to use Evernote, but I don't know if I like the program
yet.
 
Rookie uses:
In addition to what Josh listed:
  JogTracker (GPS/map based activity tracker)
  HIIT Interval Timer (Great interval timer)
  OneBusAway
  Toddler Lock (Entertainment for Seth that keeps him from messing with
buttons)
  Open Spot (mark where you find open parking spaces, look for open
spaces)
  ToDo Task Manager (decent task list with multiple widget sizes)
  WF & Clock widget (nice 2x4 clock and weather widget)