The word Aerobic means in the presence of oxygen. The prefix an- means without. So aerobic bacteria live in the presence of oxygen, while anaerobic bacteria live without the presence of oxygen.
Aerobic bacteria, in aquariums, process ammonia and nitrites. Anaerobic bacteria convert nitrate to nitrogen gas.
The thing about anaerobic bacteria and why they are not made space for in all aquarium filters is that oxygen kills them. That is why denitrators require very slow flow, so that Aerobic (with o2) bacteria can use all the oxygen by the time it gets to the media so that the Anaerobic bacteria may live.
Aerobic bacteria (I do not have the numbers for cell division) can take 4-6 weeks to establish themselves, which is what is happening when you cycle the tank.
Anaerobic bacteria, however, can take months (3-4 on a good day) to establish themselves in a denitrator because they divide slower, much slower.
Anaerobic bacteria would be killed in a wet/dry, or any environment with oxygen present. In addition to making sure that they do not have any O2, you must provide them with a constant organic carbon source (not activated carbon! Sugar, vodka, and seachem excel have been used). Too much of this source and you get hydrogen sulfide gas, deadly to your fish and to you in high enough quantities. Too little organic carbon and you get incomplete denitrification and either end up with ammonia or nitrites.