I sent this list out to a couple co-workers who were asking after they got their new Droids.
Astro File Manager - This is basically like Explorer for Android. It's a File Manager, use it, love it.
Mobile Defense (BETA) - A must have app. It is a hidden app that lets you remotely trigger an alarm, see where your phone is through GPS, wipe the device remotely. Basically a great tool if your phone is lost or stolen.
Google Voice
Amazing on this device. Seamless integration.
More Icons Widget - Free version lets you fit 3 shortcuts and the settings buttion in a 1x1 spot. The paid app lets you put 4 apps in a 1x1 spot. Basically a space saver.
iMusic - Lets you download free music based on music charts and website recommendations.
Apps Organizer - Lets you organize all your apps into categories then set shortcuts with custom icons to those app lists. (Games, Google Apps, etc.)
Dophin Browser - A must have for droid, it's a tab browser with pinch zoom. Lets you download youtube vids… pretty slick.
TV.com - Television shows streamed through 3G or Wifi to the droid. It is what it is.
Barcode Scanner - Scans barcodes, also scans the square barcodes you're likely to start seeing.
Bettercut - paid app that lets you Change the names and icons of shortcuts as well as create shortcuts to system processes.
Google Sky Map - too cool of an app. Like Google Maps only for the sky, point your phone at the stars you want to know the name of.
Nesoid - Paid app, NES emulator that works with standard NES Roms.