When BMW launched the first generation MINI in 2001, the base MINI One and Cooper models were designated R50. The Cooper S was the R53. The convertible was the R52. R53 Cooper S models were supercharged. Engines were Brazilian Tritec units, co-developed by BMW and Chrysler.
The second generation MINI, launched in 2007 grouped the Cooper and Cooper S hactchback models together as the R56, the R57 is the convertible and the R55 is the Clubman. In this generation, the Cooper S models use a turbocharger instead of a supercharger (as used in the R53). In simple terms, a supercharger and a turbocharger both do the same thing: they extract more power from the engine by force feeding the air fuel mixture into the cylinders. The difference is that a supercharger is belt driven, whereas a turbocharger uses exhaust gas to drive the compressor.
The Cooper engine in the Mk II cars (from 2007 onwards) is known as the Prince engine and was developed by BMW and PSA Peugeot Citroen.
Visually, the second generation MINI is slightly larger than its predecessor. The clearest point of visual difference is that in the second generation cars, the front indicator is positioned within the headlight, and the headlight is fixed to the body, instead of being built into the bonnet.