The essential apps that I would recommend (and that I use) are:
Advanced Task Killer - to kill those annoying apps that are written poorly for android and don't close on their own/use loads of battery (don't set it to run all the time though)
AK Notepad - simple notes app
ASTRO - file manager app
Barcode Scanner - very useful for finding apps, also pretty cool
DroidLight - see in the dark using the dual-LED
Goggles - very cool visual search function with google
Google Sky Map - fun app for finding constellations
Google Voice - adds a second number to your phone, if you have a gv account
handyCalc - graphing/scientific calculator, huge features set
Key Ring - get those dumb rewards cards off your keychain and put them on your phone
Maps - self explanatory, great for navigation, never get lost again
Movies - with rottentomatoes reviews, great for finding a movie to go to
Pandora - listen to streaming music for free
Ringdroid - create your own custom ringtones
SchottGunn and The Schwartz Unsheathed - mess around with your friends
Shazam - identify music you don't know
ShopSavvy - scan barcodes and find the best price for an item
SMS Popup - sms notification app with quick reply, reply, close delete and custom notification settings
SNesoid, NESoid and Gameboid - emulators, some of the best games for android can be played this way
Where's My Droid - never lose your phone again, text your phone a keyboard and it turns your volume up and rings if you lost your phone and it's on silent
Where - find gas stations, restaurants, yellow pages stuff etc.
Wikimobile - wikipedia for your phone