The purpose of respiration in general is to turn food into energy that a living biological cell can use. Anaerobic respiration is respiration that uses any molecule besides oxygen to do this. Many bacteria use anaerobic respiration.
Anaerobic vs. Aerobic Respiration
Aerobic respiration--which includes turning molecular oxygen into carbon dioxide--produces much more energy per unit of food than anaerobic respiration. In most parts of the world, living things that use aerobic respiration have a competitive advantage over living things that can't. However, anaerobes still dominate where oxygen levels are very low.
Facultative vs. Obligate Anaerobes
A facultative anaerobe can use aerobic respiration pathways when it has access to oxygen, and anaerobic pathways when it doesn't. An obligate anaerobe can only use anaerobic pathways, and in many cases cannot tolerate the presence of molecular oxygen in their environment at all.
All respiration was anaerobic when life on Earth first began. Photosynthesis produced oxygen as a toxic waste product until enough free molecular oxygen accumulated in the early atmosphere. This oxygen killed the vast majority of life at the time, until organisms developed systems to handle oxygen effectively and use it for aerobic respiration.