In a car or a motorbike, the engine spins very fast. The wheels need different kinds of spin — slow and strong when you start, quicker when you are already moving. The gearbox is the box of toothed wheels (gears) that changes that relationship. When you “change gear”, you are asking the gearbox to pick a different pairing. Manual cars let you choose; automatic cars have a gearbox too, but the car chooses for you. Gearbox is the usual British word, and it points quite tightly at that box of gears. American English often says transmission for the wider system. Though actually, in everyday talk people use the two as if they were the same thing. You will also hear it on bikes, buses, and some machines that are not cars at all — anywhere a set of gears sits in a case and passes power along.