The bicycle gear system that shifts a chain across multiple rear sprockets is called a derailleur.
A derailleur moves the chain laterally between sprockets or chainrings while the rider pedals. The rear derailleur also tensions the chain because the sprockets have different sizes. This lets cyclists select easier gears for climbing or harder gears for speed on flatter ground.
Modern derailleur systems use shifters, cables or electronic actuators, a chain, one or more chainrings, a cassette and front or rear derailleurs. The term comes from the French dérailleur, meaning a device that causes something to leave its track.
A derailleur is not the same as a freewheel or cassette. Those parts carry the sprockets and allow coasting, while the derailleur performs the shifting movement. Internal-gear hubs change ratios inside the hub instead of moving the chain between exposed sprockets.