A battlefest is a multiplayer mission where you start in a frigate and kill each other. Each time you die you respawn in a bigger ship. The progression is Frigate, Destroyer, Light Cruiser, Heavy Cruiser, Battlecruiser, Dreadnaught. The last dreadnaught standing wins, althou you can certainly still win if you never get to dreadnought. I can usually win against the AI when I'm still in my Heavy, sometimes Light!
Battlefests are very long, typically over an hour. With human players they can stretch to 2+. I've heard of 'fests where the combatants agree to stop to eat and run to the bathroom they go on so long!