A transmission (a.k.a. a gearbox) is a way to change the way the engine's or motor's power is used. Basically, the driveshaft connects the engine to the gearbox, and then another driveshaft connects the gearbox to the wheels. Inside a transmission, a series of gears change rpms into torques and vise verse. For more on transmissions, google 'how stuff works transmissions'.
An engine is a machine that takes the energy in a chemical (like gasoline) and turns it into mechanical energy (the stuff you need to move your car). The ICE, internal combustion engine, is the most common type of engine. The engine is sometimes referred to as the 'heart' of a car. It is where the power, measured in HP, is created. Again, more is available on howstuffworks.com
A motor is a similar to an engine because it creates mechanical energy, but it uses electricity instead of chemical energy. Most electric cars have motors.
