Browser - Firefox
Image Editing - GIMP
Webserver - Apache
Office Suite - OpenOffice (although, I don't like it... it's still the best that's free)
Operation System - Linux (KUbuntu is my preferred flavor)
Email - Thunderbird
Desktop Calculator - Qalculate
Instant Messaging - Pidgin (formerly gaim)
Vector Graphics - Inkscape
IRC client - XChat
Video Player - tie between gXine and MPlayer
CD Burning - k3b
Personal Finance - KMyMoney
LyX - in a class of it's own (WYSIWYM editor using LaTeX)
Document Layout - LaTeX
Computer Algebra System - Axiom (+TexMacs front end)
Games - SuperTux, ScummVM, Frozen-Bubble