FIDs stands for 'Falls in the Drink'. You compete for the least FIDs and try to FIDs your opponents as much as possible.
The main point of the game obviously, is that it is a duel. The most effective way of killing someone is to have them fall in the deadly boiling hot Marshmallow at the bottom of the map. This is a painful but quick way to die. As well as this you can use pickups to kill the enemy.
When someone is knocked in the drink (drink is slang for Marshmallow) by another player, the kill is awarded to the one who knocked them. If they fall, and recover, and then fall into the Marshmallow later on, then the kill will be a suicide. Knocking a player into a roll which lands them in the drink also counts as a kill. Most kills with pickups are awarded to the person who fired the pickup, but if you push someone into another person's weapon (like a weasel or puck) you will get the credit for it.
At the end of the game, the person with the least FIDs is the best player. Unfortunately, we have not yet implemented structured duels with 3 or so rounds like you would have played in the original Marshmallow Duel. This might be added in future patches if ok with the authors of various duel mutators for UT2004.