Chemiosmosis is the pumping of protons through special channels in the membranes of mitochondria. This results in a proton gradient down which protons spontaneously travel. This movement of electrons produces energy, which is used by mitochondria to bind phosphate ions to adenosine diphosphate, creating adenosine triphosphate, the main energy currency of cells.
Chemiosmosis begins with moving naked protons, or hydrogen nuclei, to an area of higher concentration, which works against the normal tendencies of diffusion. As such, energy must be used to move the proton. This energy comes from an electron transport chain. This transport chain moves electrons from electron receivers to electron donors one after the other. This movement of negative charges is attractive to the positively charged protons, which follow the chain. Once the electron transport is finished, the protons begin to move back spontaneously across the membrane toward a lower concentration.
Chemiosmosis occurs at the end of a long series of complex chemical changes that transfer chemical energy from one molecule to another, ranging from glucose all the way to ATP. If the food source is not a sugar but a fat or, especially, a protein, the process of generating ATP is even more complicated. However, this is the method used by every eukaryote on Earth.