Chemiosmosis, by definition, is the diffusion of ions through a partially/selectively permeable membrane (down an electrochemical gradient). It is specifically refers to the flow of protons through the inner mitochondrial membrane.
The main theory behind chemiosmosis is call Chemiosmosis theory, and was developed by Peter Mitchel in 1961. Here is a brief outline of the theory (now pretty much accepted as fact): Throughout respiration carrier molecules (NAD and FAD) are produced, which carry electrons. These pass electrons through to a chain of proteins which also act as electron carriers (This whole chain, from NAD and FAD to the proteins is called and electron transport chain, or ETC). These electrons release energy which is used to pump protons from the matrix through to the inner mitochondrial membrane space. The theory states that this creates an electrochemical gradient across the inner mitochondrial membrane. The protons, as of this, passively diffuse back across the membrane, down the new concentration gradient, now through protein channels attached to ATP synthase. The force of this flow (labelled proton motive force by Mitchel) provides enough energy for ADP to bind with an inorganic phosphate; driving Oxidative phosphorylation. The theory has a lot of support behind it. It successfully explains how Oxidative phosphorylation (final formation of ATP) happens better than the previous theory. Also, many studies since the theory’s conception have provided sufficient evidence to support it. For example some researchers took mitochondria and ruptured both membranes (by placing the mitochondria in low water potential solutions and detergents) to analyse where various enzymes in the mitochondria work as well as the fact that ETC enzymes are imbedded in the inner mitochondrial membrane, away from the matrix (where the link reaction and the Krebs cycle take place).
Oxidative phosphorylation did not take place without the outer membrane