My brother wants to try some runs and submit them but is confused about the rules for each of the categories. can someone please explain each of them them in detail please? Thanks!

Any% is just play through the full game start to finish.

Act 1 Any% is complete the first act from a fresh save file, with time ending when you pick up the New Game tile in the cabin.

Moon% is just kill the Moon in Leshy's phase 3. (You don't need to do the sub-quest to get the film roll like you do for Act 1 Any%, hence why this is a separate category.)

Don't worry too much if you're not sure where you're supposed to be starting/stopping the timer, the mods can retime the run for you real easily.