A transmission uses gear ratios to change the speed at which rotational energy is transferred to the wheels, which then transfer that energy to the road to move the vehicle.
Why Transmissions are Necessary
A transmission is required in most vehicles because of the limits of the engine and the range of speeds the vehicle travels at. If only the gearing needed for the top speed were used, the engine would likely stall out when starting. Using only a gear that will allow the engine to get the vehicle going would cause the engine to reach its peak running speed, and top out, not allowing the vehicle to move quickly.
Transmissions also allow the power to be transferred backwards, which allows the vehicle to be driven in reverse. Most passenger vehicles have a reverse gear, but some race cars and low-cost vehicles don't have a reverse gear.