(Keep in mind I have only been using Android since Nov 6)
Handcent SMS - A frontend for SMS/MMS. Adds a great deal of functionality to Android for messaging.
Home++ - A full home replacement that looks great and is coming along very nicely.
Movies - Find movies, theaters, and even manage your Netflix Queue.
Wifi Analyzer - A graphical WiFi analyzer.
Foursquare - Official app for the Foursquare service.
NewsRob - RSS reader with Google Reader sync.
Yelp - Official app for the Yelp service.
Seesmic - Relatively new to Android, but a fantastic Twitter client. I use Tweetdeck day to day on my laptop, but seesmic is very similar.
Advanced Task Killer - A must have app to manage running applications and even schedule app killing.
Barcode Scanner - All around barcode scanner for prodcucts or QR codes.
Evernote - Official Evernote app. Have your notes everywhere. Always synced.
DroidLive - Shoutcast internet radio player. Beautiful UI.
Remember The Milk - Official app for the best to-do list service there is, Remember The Milk.
Qik - Live stream DVD quality video from your phone.
LOLcats - Gotta get your daily fix of cheesburgerz!
Google Maps w/ Navigation - Hands down, THE BEST turn by turn navigation software I have ever used.
Weather Widget - Absolutely beautiful widget for the weather, with cool animations.
Amazon.com - Official Amazon shopping app.
Autorotate Widget - A quick one tap widget that will enable/disable the accelerometer for screen rotation.
AppsInstaller - Installer to install .apk files.
ASTRO File Manager - The best file manager I have found to date for browsing files in the Android filesystem.
Feel free to add more apps in the comments below or let me know what apps I should have picked. I will most likely have late additions since the year isn’t actually over yet. :)