Ask for a roller coaster in Paris or Madrid and you ask for "Russian mountains." Ask in Moscow and you ask for "American mountains." Each country credits the other.
The Russian claim is older. APS News and Wikipedia both place the first ice slides near St. Petersburg in the 17th century. They were tall hills of ice built purely for sliding, and riders went down on sleds. Wikipedia describes Catherine the Great ordering a summer version with wheeled carts running in grooved tracks, so the fun no longer needed winter.
The idea moved west. Russian soldiers occupying Paris in 1815 and 1816 may have brought the sledding hills with them. In July 1817 the Promenades Aériennes opened at the Parc Beaujon in Paris, with wheeled cars locked to the track. French gave the ride its lasting name, montagnes russes. Spanish and Italian use the same phrase, "Russian mountains."
The English name is American and arrived with the first big American rides. LaMarcus Adna Thompson's Switchback Railway opened at Coney Island in 1884. APS News says its top speed was 6 mph. Etymonline dates "roller coaster" to the same year. Its earliest citation is an 1884 St. Paul Pioneer-Press editorial comparing the once-novel street railway to the ride.
Both halves are ordinary words. A roller, since the 13th century, is simply a thing that rolls. "Coaster" is the agent noun of "coast," which as a verb first meant to go along the border of something, as a ship skirts a shore. Old French coste meant "rib, side, slope," from Latin costa, "rib."
Etymonline does not say which sense of coaster the ride's name had in mind. American English, though, had a winter sense ready. Etymonline records "coast" for a slide down a snowy hill by 1775, and "to coast," to sled downhill, by 1834. That is the same gravity-powered descent the Russian ice hills offered.
Russian, meanwhile, says amerikanskie gorki, "American mountains." The older Russian term was katalnaya gorka, "sliding mountain." So the ride left Russia under a Russian name and came home under an American one.
By 1959 "roller-coaster" was also a verb. Today it covers markets, moods and plot lines that rise and fall.