They would write up all the different aspects that you would go through, the different interfaces the game will have. Create naming conventions for the data (art, sound), determine what AI the enemies should have, often the plot line, and definitely how the plot will be delivered.

Basically, they are the ones who write the "screen play" of a game (design doc) which all the actors would read from to create a movie (programmers, program the game, artists draw the art, etc).

The design document should specify every aspect of the game to take out the guess work. If people know what they have to do, its a lot easier for them to get it done and move on to the next area then if they are making it up as they go along and dont have boundaries. The designer is supposed to set all those boundaries to create a "whole product" vision, which it will take a number of people to fill in the working pieces on.

-Geoff