A performance car is an automobile that is designed and constructed specifically for speed. The design and construction of a performance car involves not only providing a capable power train but also providing the handling and braking systems to support it.
Performance cars are road vehicles, capable of providing transport. Specially designed racing cars are not normally regarded as performance cars, but performance cars are often raced.
There is a great deal of overlap between performance cars and sports cars, but not all performance cars are sports cars, and not all sports cars are performance cars. The MG TC, for example, is a prototypical sports car, but even by the standards of its day was unexceptional in its performance.
Similarly, some hot rods are performance cars, but many hot rods are either built for show rather than for performance, or are not intended to be roadworthy, or both. Those hot rods that are built for on-road performance tend to emphasise straight line speed and acceleration, and in this they represent one extreme of a continuum of performance cars, the opposite extreme of which is represented by cars that emphasise handling rather than straight line speed, such as the Lotus Elise.