Supermoto basically is the term for the multi disclipine format racing that people started doing back in the late 80’s early 90’s where they wanted to get the best of both moto-x and closed course street racing… Usually Supermoto refers to this kind of asphalt circuit that includes a dirt section with jumps that makes up about 1/4 to 1/3 of the track…[EDIT: I see this has already been mentioned by Mingshah who posted while I was writing this 
…] Supermotard usually is used to refer to all asphalt closed course racing using the same type of bikes… think go kart tracks… these aren’t hard and fast rules at all, but generally speaking that’s the terminology…
MotorcycleRider’s already explained the differences in wheels and tires between Moto-X and Supermoto bikes, basically 17" spoked rims front and rear and sportsbike sticky tires instead of the larger 19" or 21" front wheel size found on most Moto-X and Enduro bikes…
The cornering / lean angle story is that Supermoto’s “gimmick” (kind of the wrong word since it’s totally legitimate) is the foot out, bolt upright, slide the rear into the corners similar to dirt tracking and flat tracking… basically you let the bike lean under you, instead of hanging off racer style because you want to break traction and slide the rear out… and that way if it grips you can control the impending high side much better with one foot on the ground and your weight off the bike as much as possible than you could hanging off GP racer style… backing the rear into corners supermoto style actually makes MotoGP tire sliding action seem pretty tame, although that’s only if you ignore the fact that supermoto’s are not doing it at 200km/h+ every corner… supermotos however get insane lean angles and often seem to be in danger of getting the handle bar down 
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@ hm: my turn to correct you today… according to BMW, GS actually stands for Gel