Mitochondria make energy for cells. They do this by way of oxidative phosphorylization. ( Oxidation is the loss of Hydrogen atoms, phosphorylation is the attachment of phosphate group to something else. )
Proteins imbedded in the inner membrane of the mitochondria move H+ from the matrix to inner membrane space and back again. This allows there to be more electrons in the inner membrane space than in the matrix. This difference allows electrons to flow back through the membrane via the ATP synthase enzyme.
ATP synthase moves electrons into the matrix . Somehow ( sorry, don't quite understand this step ) this allows for the creation of ATP.
ATP leaves the mitochondria, encounters the cell membrane where ATPases grap hold of one of the phosphate groups and snap it off releasing heat and energy, thus creating energy for the cell.