A chauffeur is, literally, the person who keeps the fire going.
Etymonline says the English word dates from 1896, when it meant "a motorist." It came from French, where it meant "stoker," the person who fed fuel to a steam engine. The French noun comes from chauffer, "to heat." The French used the word as a nickname for early motorists, Etymonline explains, because the first motor cars were steam-powered.
Merriam-Webster tells the same story. The first chauffeurs were hired to stoke and maintain a steam engine, and French speakers later applied the word to people who drove early automobiles. Dictionaries differ on the first English date. Merriam-Webster gives 1899, while Etymonline quotes an 1896 Horseless Age article saying that Charles B. King "acted as 'chauffeur,' as the French say." Both agree the word came from French.
The verb behind it has cousins in English. Merriam-Webster and Etymonline both note that chauffer goes back to Old French chaufer, which also gave English the word chafe. So a chauffeur and a chafed hand share an ancestor in the idea of heat.
The meaning narrowed in the early twentieth century. Etymonline records the sense of a paid or professional driver of a private car in 1902, along with a verb meaning to drive as a chauffeur. It also notes that early twentieth-century British English had a jocular anglicized form, shover.
Cars soon stopped needing anyone to tend a fire. The fire was gone, but the job title had settled into the language. Today a chauffeur is someone paid to drive, and nobody expects them to shovel anything.