AMG (Aufrecht Melcher Grossaspach) was the leading independent tuner of Mercedes-Benz automobiles when they were purchased by the company in the mid 1990’s. AMG vehicles are meant to be the highest performing model for a specific model line (though not necessarily the most luxurious). They have handbuilt motors, upgraded brakes and suspension, and exclusive interior and exterior featuers. Since they are factory built they are covered by a manufacturer’s warranty. They are also produced in relatively limited numbers. They can be identified by the badge on the trunk, as well as the two digit model designation. (i.e. C32 vs C320)
Simplest answer: They have bigger engines. Each one is has based off of one of their current models (like the C-Class – C300, C350), but then they put in a bigger engine and make the styling more race-like (the C63). I think they usually have the letter of the model, but only two numbers following (usually based on the engine size, # of liters). The C63 though is really 6.2L, but the 6.3 was to pay homage to some original AMG I think.