GT300 is the category that private teams compete in, and is composed of cars in three different rulesets, all balanced together for fair competition - JAF GT300, FIA GT3, and Mother Chassis.
Mother Chassis is based on a spec chassis (that supports both front- and mid-engine cars), and a spec engine (a 4.5 liter V8). You put whatever body you want on that chassis.
FIA GT3 is a popular spec used worldwide for rich people that want to buy a race car. Meh.