Aerobic respiration requires oxygen, anaerobic respiration does not.
In aerobic respiration glucose and oxygen react together in cells to produce carbon dioxide and water and energy. In particular this occurs in mitochondria of cells.
In anaerobic respiration glucose is broken down incompletely to produce lactic acid and energy relatively quickly. In particular this occurs in the cytoplasm of cells.
Oxygen debt is the additional oxygen that must be taken into the body after exercise, to restore all systems to their normal states by oxidising the lactic acid.