The subsequent movement of ions from an area of higher concentration to an area of lower concentration through transport proteins on the selectively-permeable membrane as a result of a proton gradient that forms across the membrane that is not readily permeable to ions.
More specifically, chemiosmosis is the diffusion of hydrogen ions (protons) across the biological membrane via the ATP synthase (a transport protein) due to a proton gradient that forms on the other side of the membrane. The proton gradient forms when the hydrogen ions accumulate as they are forcibly moved to the other side of the membrane by carrier proteins while the electrons pass through the electron transport chain in the membrane. Since more hydrogen ions are on the other side they tend to move back across the membrane via the ATP synthase. As they flow through energy is released which is then used to convert ADP to ATP (by a process called phosphorylation).
Chemiosmosis is one of the processes by which ATP is synthesized. In eukaryotes, it takes place in the mitochondria during cellular respiration and in the chloroplasts during photosynthesis. In prokaryotes, it occurs in the cell membrane.
This process is called chemiosmosis because the chemical ions move from an area of higher concentration to an area of lower concentration across a semipermeable membrane, similar to the movement of water molecules by osmosis.
See also: chemiosmotic theory