To make things simple, a muscle car is basically a mid or full size 2-door car with a large displacement (300 Ci plus) v8 in the front powering the rear wheels. They are usually heavy, pack a lot of torque and are good for (well, were good for) acceleration and straight line speed. They rose to prominence during the period late 1950's upto the late 70's and were used widely in drag racing, both legal and illegal. Movies such as "Vanishing Point" and "Bullit" helped in broadening the Muscle car culture which lasted for quite some years til the Fuel Crisis.
Due to the fact that the products were positioned in the favor of Power rather than handling, the cars were terrible around corners but it wasn't a big problem as America mostly had huge straight roads with barely any turns. This neglect for better handling saved costs and helped keep the prices low hence, muscle cars were a huge hit amongst the youth of the 60's and 70's.
The term "Muscle Car" was coined by the Americans but over the years, there have been variations in the definition of a muscle car. Nowadays, many manufactures refer to their own cars as Muscle cars, such as the Mercedes C63 AMG due to it's monstrous torque figures and the Holden Monaro which is the Australian variant of the Pontiac GTO.