Here is my list of android apps that I find most useful:

Advanced task killer – kills all apps that running. helps save memory and battery life

aTrackdog – shows you what apps need to be updated

Barcode scanner – scans bar codes which allows you to download apps and scan bar codes at stores

CardioTrainer – records you walks, runs, and bike rides distance, time, and map

Dropbox – can be used to share files with your desktop

Gdocs – gives you access to your google docs

ringdroid – lets you create your own ring tones from mp3s

RealCalc – a scientific calculator

Google voice – sends me my email via text so i dont have to listen to it.

Additionally I use:

accuweather – weather app and widget
adobe reader – read pdfs
antivirus – protects phone from viruses
coupons – has coupons for stores and even has barcode scanner for register
documents to go – reads word and excel files
goggles – takes a picture of a place and shows you where it is
kindle – lets u read kindle books
places directory – finds restaurants, gas stations, etc
shazam – if a song is on the radio it will tell me what its called and who sings it
voice recorder – makes droid a voice recorder
yellowbook – to look people up
zillow – to look up cost of homes anywhere
pandora – internet radio

of course there are apps that i use a lot that i did not list such as facebook, twitter, gmail, google maps with gps, etc but these are common and actually came with my android 2.1 OS. As far as widgets go, I only have one because they drain a lot of battery. The one i use is the power widget, which allows you to turn power settings on/off if you are using them (wifi, bluetooth, gps, sync, brightness).

All of the apps I have mentioned are free!