The movement of ions across the membrane depends on a combination of two factors:
# Diffusion force caused by a concentration gradient - all particles tend to diffuse from higher concentration to lower.
# Electrostatic force caused by electrical potential gradient - cations like protons H tend to diffuse down the electrical potential, from the positive (P) side of the membrane to the negative (N) side. Anions diffuse spontaneously in the opposite direction.
These two gradients taken together can be expressed as an electrochemical gradient.
Lipid bilayers of biological membranes, however, are barriers for ions. This is why energy can be stored as a combination of these two gradients across the membrane. Only special membrane proteins like ion channels can sometimes allow ions to move across the membrane (see also: Membrane transport). In the chemiosmotic hypothesis a transmembrane ATP synthases is central to convert energy of spontaneous flow of protons through them into chemical energy of ATP bonds.
Hence researchers created the term proton-motive force (PMF), derived from the electrochemical gradient mentioned earlier. It can be described as the measure of the potential energy stored (chemiosmotic potential) as a combination of proton and voltage (electrical potential) gradients across a membrane. The electrical gradient is a consequence of the charge separation across the membrane (when the protons H move without a counterion, such as chloride Cl).
In most cases the proton-motive force is generated by an electron transport chain which acts as a proton pump, using the Gibbs free energy of redox reactions to pump protons (hydrogen ions) out across the membrane, separating the charge across the membrane. In mitochondria, energy released by the electron transport chain is used to move protons from the mitochondrial matrix (N side) to the intermembrane space (P side). Moving the protons out of the mitochondrion creates a lower concentration of positively charged protons inside it, resulting in excess negative charge on the inside of the membrane. The electrical potential gradient is about -170 mV , negative inside (N). These gradients - charge difference and the proton concentration difference both create a combined electrochemical gradient across the membrane, often expressed as the proton-motive force (PMF). In mitochondria, the PMF is almost entirely made up of the electrical component but in chloroplasts the PMF is made up mostly of the pH gradient because the charge of protons H is neutralized by the movement of Cl and other anions.  In either case, the PMF needs to be greater than about 460 mV (45 kJ/mol) for the ATP synthase to be able to make ATP.