A stud is a 2x4 or something similar that is in the wall behind the drywall. Generally you can't visually tell where they are. Sometimes you can tell by knocking and listening for a sound difference. You can also buy a stud finder that will sense a difference in density and let you know where it is.
As for it being necessary to mount the full metal bracket on a stud, I am not sure. The prudent thing would be to use a stud. I also wonder if some sort of drywall anchor would work, but I have no idea.
-Sean
