The transmission is a drivetrain component in your vehicle that transmits energy from your engine to the drive wheels. In short, it is a large, metal gearbox that sends rotational power through a drive shaft to another, smaller gearbox called a differential. From there, that power is transmitted through axle shafts to the wheels.
Much like shifting gears on a bicycle, a transmission changes gears to allow your engine to operate with the most effective amount of power to match the speed you are traveling. Some transmissions are manual versions (where you shift the gears manually) and others - the majority of models on the road today - are automatic versions (where a computer control module determines when the transmission should shift). The transmission on a front-wheel-drive vehicle is known as a transaxle, as it serves the purpose of both transmission and differential. Some transmissions, known as Continuously Variable Transmissions (CVTs), rely on a pair of internal pulleys connected by a belt to match engine speed to travel speed, (similar to the way a snowmobile works) rather than using gears.
Transmission replacement is a service that involves removing your old transmission and replacing it with a new one (although it can alternatively be replaced with a rebuilt, remanufactured, or pre-owned unit).