The halacha states that nails have to be clean (no dirt underneath) and without hangnails. The custom is to shorten them in order to make cleaning easier. It is sufficient to cut your nails short enough so that if you look at your hand from the palm side you don't see them. You certainly should not cut them so short that it hurts. If there is a need for longer nails (e.g., for guitar playing or sewing), there is room for leniency to allow longer nails as long as they are well cleaned prior to mikveh use.