A (usually) North American version of the European English 'Naught', meaning nothing, although in the case of aught, it is primarily used in the numeric case. Most commonly used in dates ('Back in aught-four...') or in weaponry (The Thirty-aught-six, though the spelling "Thirty-ought-six" is much more common).
This word is seldom used by young people, but it stands a chance of being put back into fashion due to it now being the 21st Century, and we can start using 'aught' again.