Bioremediation is the use of living organisms to clean up a pollutant in a natural setting. Biodegradation is basically another term for this. It is the chemical breakdown of unwanted waste materials using living organisms. Usually these living organisms are microbes. Aerobic and anaerobic refers to the presence or absence of oxygen during the chemical breakdown process. An aerobic organism functions in the presence of oxygen. An anaerobic organism functions without the use of oxygen.
One particularly useful area of aerobic bioremediation is the clean up of spilled oil in ocean water. One of the worst aspects environmentally of spilled oil are polycyclic aromatic hydrocarbons (PAH's). These have notoriously long lifetimes and tend to bio-accumulate (in other words, build up over time in an exposed organism's tissues). A new strain of the bacteria Pseudomonas aeruginosa (called NY3) helps to break down these PAH's in oil spills by oxidizing the carbon into harmless byproducts. This is an example of aerobic bioremediation.
An example of anaerobic bioremediation is Dechloromonas aromatica. This is a microbe that can break down benzene in an anaerobic environment through a mechanism that has not even been fully elucidated yet. Benzene is a very stable known carcinogen than is present in oil/petroleum products and is a byproduct of some industrial processes.
Some chemical pollutants that need to be cleaned up include oil/petroleum that spills both on land and water. Also, excessive nitrogen and phosphates from fertilizer and agricultural use also tend to accumulate and can be treated with bacteria to break them down. Manure and livestock waste can contaminate waterways if allowed to runoff from pastures. Bacterial treatment can break down this waste before it can become a danger to humans through drinking water.