Basic definition of the category can be found on: What is a Light-Sport Aircraft? on sportpilot.org.
Licence needed for LSA flying
Pilot must own a sport pilot licence, which has following restrictions 1:
cant carry more than one passenger
VFR day - daytime flight only (civil twilight is used to define day/night)
cant fly above 10,000 feet (3,000 m) MSL (sea level height) or 2,000 feet (610 m) AGL (above ground level), whichever is higher
no flight in any of the airspace classes that require radio communication (A, B, C, D) without first obtaining additional instruction and instructor endorsement
FAA certified LSA models
There is a list of FAA certified models.
References
[1] Pilot certification in the United States
[2] Sport pilot rule synopsis
MTOW: max. 1320lbs (600 kg)
Stall speed: max. 51 mph (75 km/h) clean (without use of flaps, can have flaps though)
Single normally apirated engine
Fixed propeller
Fixed landing gear
Max. cruising speed: 138 mph
These airplanes might be Light sport aircraft - LSA eligible