Despite what many people believe, the Wankel Rotary Combustion Engine (Wankel Engine) is an internal combustion engine, just as the more widely used piston engines (boingers to Rotary enthusiasts) are. Also like a conventional piston engine, it is a Four Phase cycle engine, with 4 clear phases: intake, compression, power, exhaust. This is also referred to as an Otto Cycle Engine - or 'Four Stroke'.
How it does these things is how it varies. The engine itself spins, like a turbine (and I believe in a manner similar to Tesla's engine, but that is open to dispute!). There are only 3 main moving parts in most Wankel engines: 2 rotors and the eccentric shaft which rotate one way continuously; there is no piston like motion as in a conventional boinger. You can see how it operates in the image above.
Because of the method of moving the fuel air mixture, the four phases take place in different areas of the engine while the mixture itself eddies within. This turns out to be an advantage for Hydrogen fuel, and is a reason that hydrogen fueled car experiments usually entailed a Wankel engine.
Compression in a rotary engine is created by the rotor and the engine peripheral housing (engine 'housing') - if you look carefully at the animated image above, you'll note that the gas expands within the SAME space that it was fed into - the compression of the burning gas once ignited spins the engine until it is blown from the exhaust.
The shape of the housing can be seen if you watch the the animation as well - watch where the rotor itself comes
into contact with the housing. This is called an Epitrochoid Curve, and the mathematics behind it is something I
hope to put up here as soon as possible. 
It should be noted that the technology associated with the rotary engine predates most of the technology
associated with the piston engine:
1) The Rotor predates the piston engine's reciprocating piston.
2) The Eccentric Shaft predates the piston engine's crankshaft and connecting rods.
3) The Peripheral Housing supercedes the piston engine's cylinder.
4) Intake and Exhaust Ports in the housings eliminate valves, camshafts, cams, lifter rods, and timing belts.
Wankel engines use cycloidal gearing, an old and unusual form of gearing used in mechanical watches and screw compressors
