For the Phone
Backup Everything - where most other backup apps just backs up one type of item, eg applications only, or SMS only
TasKiller - to manage phone memory by forcefully killing idle apps that still hog memory after you've closed them
Ultra Keyboard (non-free) - to support special characters like ; | and ALT CTRL etc we'll need them for SSH
Remote Access
SSH - ConnectBot
VNC Client - VNC Client, for remote desktop access to VNCServer-enabled remote computers/servers
VPN Client (part of Android)
Servers
Web Server - kWS http server
FTP Server - FTPServer
Instant Messengers
Skype for Android - pretty good if you have a short list of contacts. I have way too many contacts, way too many groups that I uninstalled this after trying it for awhile. The messages just keep going and going to the point it sucked my battery like a leech! Otherwise works well on text and voice chat subject to Skype's server bandwidth and available internet bandwidth on the phone
WhatsApp - it's great for occasional text contact with others especially those friends who are using iPhones, since this seems to be all the rage for iPhone users (not sure why). Will keep it for as long as it is free (only first year for Android) then will decide then whether to pay or ditch it for good.
GTalk - native app, v light footprint, v handy if everyone was using GTalk at work/home. Works well on text and voice chat.
Social
Facebook! who can live without this app nowadays... on the phone it only doesn't support video uploads yet, otherwise it's almost as good as the web version if not better.
I don't tweet so I won't be able to comment about the Twitter apps on Android, but I would expect them to be pretty decent too.
Multimedia
GTunes - music player + allows preview/download of tunes from around the world, allows searching of popular tunes and listing by popular music charts eg American/UK/Japan music charts
tTorrent - BitTorrent client for Android! Allows torrent searches and multiple torrents, turn on/off if not on WIFI, very light memory+cpu footprint
Other Productivity Tools
Email - GMail (default app in Android) - super tight and seamless integration with Google Apps Email
Contacts - (default app) very tight integration with Google contacts as well as Facebook contact information
Task Manager - GTasks - Android app for Google Apps' Tasks
Document Viewers - QuickOffice for Microsoft Office docs, Adobe PDF Viewer, URLtoPDF - allows saving of webpages to a PDF file saved in your phone
Notes - Evernote - free account gives a generous amount of space on Evernote servers, allows syncing of your notes (text+media notes) on all your devices
File Manager - ES File Manager - allows direct management of files and folders on your phone and SDCard, comes with its own image/video/file viewer
WordPress - I use this to take snapshots of cheques (payments) received from clients and upload as a private post. Helps me keep tabs on my accounting.
Drupal Editor - I use this to update my blog site at azhan.net, but now need to check if there is an update to support Drupal 7 :) this one was previously working well on Drupal 6 - it just works!
Navigation
Google Maps - extremely useful for finding places, getting directions from A to B etc and to some extent traffic conditions
Compass - nothing like having a compass on your phone when you need to recalibrate your bearings
Other Handy Apps
Flashlight - HTC brand flashlight, with OFF and 3 brightness levels when ON. The on-screen flashlight even looks pretty cool ;)
Currency Converter - FX - handy tool to give you an idea of foreign currency rates when on the go, supports most currencies
Stocks - allows monitoring of stock prices, (source: Yahoo!)
Favourite Frequencies - main use for this is to turn on the 17MHz frequency to ward off mosquitoes when I am spending nights in Malaysia or other places known to have these pests lurking around the bedroom