Two broad categories of microorganisms consume and decompose organic matter: those that need air (aerobic) and those that don't (anaerobic). Most folks who compost rely on aerobic, aboveground decomposition. It's the simplest method to start with because all that's required is a pile of organic matter.
Aerobic composting
Aerobic composting is the principle at work in aboveground composting environments — whether it takes place in a freestanding pile or in a container that provides air circulation, such as a bin with open sides or a tumbler with aeration holes.
As long as plenty of air is available, aerobic decomposers work faster and more efficiently than their anaerobic counterparts, providing you with finished compost on a faster timetable. However, as organisms deplete the supply of oxygen from the existing spaces and pores between bits of organic matter, the decomposition process slows.
To keep your decomposers working at maximum speed, you may want to incorporate some type of aeration aid during your initial pile construction. One way to do this is to pile organic materials on top of a recycled shipping pallet. The pallet sits several inches above the ground's surface, allowing air to flow beneath it.
If you notice your compost pile shrinking, you can reenergize your aerobicizers by giving your pile a fresh infusion of oxygen in a couple ways:
Turn your pile completely: Fork a freestanding heap to an adjacent spot or turn the contents of one bin into another. If using a tumbler, give it a spin.
Stir organic matter regularly: Use a pitchfork or an aerating tool to stir things up.
If your compost is emitting a bad odor, like rotten eggs or ammonia, it's too wet or wasn't thoroughly mixed. A well-constructed compost pile doesn't smell bad. In fact, it emits a refreshing earthy aroma, like kicking up leaves during a walk through the woods.
Aboveground aerobic decomposers can withstand higher temperatures than their anaerobic counterparts, and they generate heat as a byproduct of their activity. Not all aboveground piles are "hot," but when conditions are to the decomposers' liking, temperatures in your pile heat up sufficiently to kill weed seeds and pathogens.
Anaerobic decomposition
Anaerobic organisms work without oxygen, so most anaerobic takes place underground in pits or trenches. Basically, you dig a hole, fill it with organic matter, and seal it with a layer of soil. Anaerobic decomposers get right to work, with no need for fresh O2.
Anaerobic organisms work at slower rates than their aerobic counterparts, and it's impossible to monitor their progress without digging into the hole and poking around. Anaerobic organisms exude smelly gas as a byproduct of their exertions. And because of the colder conditions, weed seeds and plant pathogens aren't destroyed.
Despite these disadvantages, anaerobic composting is the best way to go in some situations:
You're looking to dispose of a one-time load of wet, potentially smelly, or pest-attracting kitchen waste, such as you'd accumulate after a day spent canning fruits or vegetables, cleaning freshly caught fish, or organizing a big social gathering that generates food scraps.
Pulling spent garden plants at the end of fall leaves you with an enormous pile of organic matter that you don't have the space or time to manage over winter.
Aboveground composting of kitchen scraps without a sealed container isn't allowed where you live.
You aren't keen on the appearance of a compost area in your landscape, but you prefer not to send your organic waste to a landfill.
You want to improve soil structure and fertility in a future garden bed.
You don't have time to monitor the air or moisture requirements of an aboveground compost pile.