What are the functions of ATP synthase?

ATP synthase is a protein complex found in the cristae of the mitochondria in eukaryotic cells carrying out aerobic respiration.  Its purpose is to generate ATP from ADP and inorganic phosphate. This is accomplished by coupling the electron transport chain and its energy release to produce the ATP by a process known as chemiosmosis.

ATP synthase uses energy of an ion gradient to power production of ATP. It can do this because there is...

ATP synthase is a protein complex found in the cristae of the mitochondria in eukaryotic cells carrying out aerobic respiration.  Its purpose is to generate ATP from ADP and inorganic phosphate. This is accomplished by coupling the electron transport chain and its energy release to produce the ATP by a process known as chemiosmosis.

ATP synthase uses energy of an ion gradient to power production of ATP. It can do this because there is a difference in concentration (gradient) of H+ ions on opposite sides of the inner membrane of the mitochondria, which is also a pH gradient as well. 

When a gradient exists across the membrane, as the ions fall down their concentration gradient, the release of energy can be used to do work including synthesis of ATP. The electron transport chain in respiration uses the energy from the flow of electrons which is exergonic to pump H+ ions across the membrane from the mitochondrial matrix to the intermembrane space, from low to high concentration. These ions must flow down their gradient and the only way to get across the membrane  and back to the matrix, is through the enzyme ATP synthase.

As protons move onto binding sites on ATP synthase in a place called the rotor, it causes it to spin which causes the ADP and inorganic phosphate to join to form ATP.

In a sense, the ATP synthase is a tiny motor within the membrane. 

This is an example of chemiosmosis  which couples the energy from a proton gradient to drive cellular work including ATP synthesis.  ATP synthase can also be found in the cell membrane of prokaryotes, and in the thylakoid membrane of producers that contain chloroplasts. Its purpose is to catalyze the formation of ATP from ADP and inorganic phosphate. 

I have included a link with a diagram of ATP synthase structure.