In mechanics, a transmission or gearbox means gears working together. The most common transmission is the transmission in a car.
In a car, the engine makes the crankshaft spin. The crankshaft makes the transmission's gears spin, which spin some more parts, which make the wheels spin. But the wheels do not spin as fast as the engine makes the crankshaft spin. That is because the gears are made to slow down the spinning. The gears also give the spinning more force (or to be more scientifically exact, more torque), so the wheels can make the entire car move.
A transmission has some gears that change the torque more than other gears. That is why a transmission has a first gear, second gear, third gear, and sometimes more gears. There is also a reverse gear and neutral. An automatic transmission shifts between gears by itself (except reverse gear, and a parking gear). A manual transmission needs the driver to decide when to shift gears. Manual transmission is also called standard transmission. This is confusing because almost all cars in the U.S. have automatic transmission. In other words, "standard" transmission is unusual in the U.S., not standard.
Not all transmissions are in cars. For example, bicycles have transmissions. You can see the gears, so it is easier to see how it works.