The first robots were not made of metal, and the word for them came from the playwright's brother.
In 1920 the Czech writer Karel Čapek wrote R.U.R., short for Rossum's Universal Robots. It is about a factory that mass-produces artificial workers. Wikipedia notes that his robots are living, biological beings, assembled from parts rather than born. The play describes vats for liver and brains. They are closer to what science fiction now calls androids.
Čapek first had another name in mind. Science Friday reports that his early drafts called the creatures labori, from the Latin root for labor. He worried it sounded too bookish. His brother Josef suggested roboti instead. Karel later credited Josef in print as the true inventor of the word.
Josef's choice carried a sharp meaning. Czech robota meant "forced labor" or "compulsory service." Science Friday links it to Central European feudal systems, where tenants paid rent through forced labor. Etymonline traces it to Old Church Slavonic rabota, "servitude," and rabu, "slave." A robot, in the original sense, is a serf.
The play premiered in Hradec Králové on 2 January 1921, then at the National Theatre in Prague on 25 January. It crossed the Atlantic fast. A New York production opened in October 1922 and ran for 184 performances. London followed in April 1923. Etymonline dates robot in English to 1923.
The word then escaped the play and became the ordinary name for artificial workers. Etymonline records a figurative sense too: a person whose work is entirely mechanical.
The root reaches further than Czech. Etymonline traces the Slavonic words to a Proto-Indo-European root, orbh-, meaning "to pass from one status to another." The same root gave English orphan, a child whose status has changed by losing its parents. It also gave German Arbeit, "work." So the robot, the orphan and the German working day all share one ancient ancestor.