The mitochondria
Often the mitochondria is claimed to be the powerhouse of a cell. It appears in both plant and animal cells and is the organelle that carries out cellular respiration.
Normally the mitochondria consumes oxygen and glucose to produce carbon dioxide and water as waste products (used in to complete the cycle) and produces heat. Wikipedia has both the equation of
##C_6H_12O_(6(s)) + 6O_(2(g)) -> 6CO_(2(g)) + 6H_2O_((l)) + heat##
as well as
##DeltaG = 2880 kJ/mol## of ##C_6H_12O_6##
Essentially the mitochondria releases energy for bodily functions when it breaks apart glucose.
Without oxygen however the Krebs Cycle cannot be completed but Glycolysis still can be. If the body still needs to produce energy but lacks the oxygen for the Krebs Cycle pyruvate acid is not metabolized. It then resorts to fermentation.
Fermentation is a process where significantly less ATP is produced compared to regular cellular respiration. This is what causes lactic acid to build up in us humans if we exert more energy than the oxygen we have and are taking in can help supply. This is what can cause muscles to become sore for a while.
However the mitochondria can still function perfectly fine in some organisms. Some microorganisms use anaerobic respiration while we use as well as most organisms. Anaerobic respiration does not require oxygen. Instead it will use an inorganic acceptor such as a sulfate or nitrate.