How does a Compression ignition engine work?

How does a Compression ignition engine work?
A compression ignition engine works with Pistons and Valve openings.
There is an intake valve, with lets the fuel and air into the cylinder, and the exhaust valve which allows for the escape of the combustion.

The engine starter, supposedly started by a battery, gives the piston an Initial movement away from the valves, which bring the fuel and air into the cylinder, then pushes the piston back up, with this mixture of fuel, which is at a certain mixture which is then ignited by a spark plug. The explosion throws the piston back down which releases the exhaust valve allowing for the exhaust to exit. Once the exhaust leaves the piston moves back up , probably by the flywheel... and is brought down again which allows another mixture of fuel. This allows the engine to begin on its own timing without the need of a starter.