The transmission takes the rotational power of the engine,or torque, and converts it to a more useful form of power by way of gear ratios. The two main types of transmissions are manual and automatic. The transmission is bolted to the back of the engine. It is mechanically connected in one of two ways,depending on the type of transmission it is.
Manual
A manual transmission is connected to the engine flywheel by a clutch plate and pressure plate. When the clutch pedal is pressed,the pressure plate is pulled away from the clutch disc and interrupts the power flow of the engine to transmission. This action releases the clutch from the turning engine. This is how you are able to shift gears.
Changing gears is performed by moving the shifter lever. This shifter lever engages and disengages different sets of gears on a shaft. The different sizes,or ratios,of the gears are what provide the correct power output.
Automatic
In an automatic transmission,the torque converter is the mechanical connection between the engine and transmission. The primary function of the torque converter is to transfer the power from the engine to transmission using hydraulic (transmission) fluid. The fluid coupling action of the torque converter is what allows the engine to stay running when stopped with the vehicle in a forward or reverse gear. Gears are changed automatically by hydraulic pressures and solenoids that either hold or apply bands,servos,and gear sets.
Read more about how to tell if a torque converter is bad here.