The big idea
Gears trade force for distance. A big chainring and a small cog turn the wheel many times for each turn of the pedals: fast, but hard to push. A small chainring and a big cog turn it fewer times: slow, but easy, so you can climb a steep hill. A chain carries the pull from the pedals to the wheel, and derailleurs lift it from one sprocket to the next.
Memory trick: “Big in front, small in back: fast. Small in front, big in back: climb.”
A bit of history: In 1880 Hans Renold, a Swiss-born engineer working in Salford, near Manchester, patented the bush roller chain, the kind of chain bicycles still use. In 1885 John Kemp Starley of Coventry showed his Rover “safety” bicycle, which drove the rear wheel with a chain, so both wheels could be about the same size. In 1949 the Italian racer Tullio Campagnolo showed the Gran Sport, a parallelogram rear derailleur that went into production soon after. Nearly every rear derailleur since works the same way.