Every week, there is a challenge on the Brogue Forums to complete a game with a particular "seed". The seed is a number used by the random generator to kick off a sequence of numbers that are then used to generate the dungeon. Everyone who plays the same seed game will see the same dungeon, monsters, and items in the same order and in the same places. They will, however, get different combat rolls -- while both start the same, they quickly diverge as players play. (Two players who chose exactly the same sequence of moves would see exactly the same outcomes; this is the mechanism behind Brogue's save and recording system.)
To play a seed game, hold down the control button on the main menu screen. "New Game" will turn into "New Game (custom)"; clicking it will prompt you for the seed. On the Linux build, you can use -s number to pick a seed from the command line.
Valid seeds are from 1 to 4,294,967,295. However, when you click "new game" without specifying a seed, Brogue almost always generates a seed over 1,000,000,000 (because the seed is based on the system time). If you use the Seedscummer to look for a seed matching particular requirements, it will start looking from a seed much less than 1,000,000,000. If you enter a number over 4,294,967,295, seed number 4,294,967,295 will be generated.
You can see the seed of your current game by pressing tilde ("~"). If you find a particularly generous, difficult or interesting seed you can share it by posting it on the forums.