Until the Krebs cycle, aerobic respiration can be explained without mentioning oxygen, the chemical element after which the reaction gets its name. Where in the process does this chemical element take part? What is its importance?

Oxygen enters the aerobic respiration in its final phase, the respiratory chain. It is of fundamental significance because it is responsible for the maintenance of the hydrogen concentration gradient among the spaces separated by the inner mitochondrial membrane. This gradient promotes the functioning of the ATP synthetase and therefore the phosphorylation of ADP to form ATP. In the space inside the inner membrane oxygen binds to free hydrogens to form water and this hydrogen consumption keeps the hydrogen gradient and the proton traffic by the ATP synthetase.

The whole aerobic respiration process has the intent to make the ATP synthetase work. Aerobic beings, for instance, we humans, need to breathe oxygen to maintain that hydrogen concentration gradient and keep the ATP synthetase working.

Cell Respiration Review - Image Diversity: ATP synthetase