Blimey - PPRuNe at it's best. A simple question asked by someone seeking enlightenment and they get a thesis and a sh!t fight!
I would like to answer the original queries by by answering the second question first. In simple terms, the power output of a reciprocating engine can be expressed as bore x stroke x rpm x compression ratio. Based on this, if you force more of a charge into the engine by using a turbo or superchager, the compression ratio will increase and the power output will go up.
As for the first part, obviously you understand the OTTO cycle an the principle of induction, compression, conmbustion and exhaust, or suck, squeeze bang and blow. The turbojet engine does the same. The suck is the flow into the engine, the squeeze comes from the compressor, the bang occurs in the combustion chamber and the blow is the thurst. If you look at it like this, if you didn't have the compressor, the engine wouldn't work and as someone else said, all you would have is a bonfire!