Anything which is being compressed is in a state of compression. Nearly everything on Earth is being compressed in some way, but in an engine, the idea of compression is usually referring to how much the piston squeezes the air in the combustion chamber. This is often measured as a compression ratio. In a diesel engine, compression is used to ignite diesel fuel without the use of a spark plug.
In gasoline engines, too much compression can cause detonation, and how much compression is required to cause this depends on the octane rating of the fuel (higher octane fuels are better at resisting detonation).
Compression levels can be increased by forced induction methods such as turbochargers and superchargers, but these do not alter the compression ratio, which is based on the amount of displacement caused by the piston within the cylinders of the engine. The ratio is unaffected by the starting pressure.