(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. 🙂