Almost all motor bikes use Sequential transmission . This type of transmission is similar to a manual transmission in car . This system is used because the amount of area available is less .The method of engagement mechanism is different from those in cars.
Its working is hard to be stated in words just check out ths video .
The final drive is a chain drive.
Transmission in Motor scooters
Motor scooters use a kind of automatic transmission called as CVT ( continuously variable transmission).this kind of transmission is also used in atv and cars.
This systems consists of two seamless tapered cylindrical surfaces engnaging together,mostly a belt is used to transmit power between these two.
Variable gear ratio are created by sliding contact between belt and surface of cones
The motion between pulley is controlled , component called a variator.The variator function is similar to a centrifugal clutch.