"Aerobic" pertains to an organism that needs atmospheric oxygen to thrive. In landscaping and gardening it is used in reference to compost bins. Well-designed compost bins that are well maintained promote an environment in which such aerobic microorganisms thrive (unless you've opted for the opposite approach, which is anaerobic composting).
Aerobic organisms are an important ingredient in good compost; without them, the material in a compost bin won't "cook" properly. If you've stuck your hand into a compost bin and found it hot, then aerobic organisms were present in sufficient numbers in that pile.
The process of aerobic composting entails applying the correct mix of organic materials, applying the right amount of water, and turning the pile occasionally with a pitchfork. The right combination of these elements will put aerobic microorganisms to work for you, breaking down the organic materials in the pile.
For more information, see the full article on how to make compost.
