Roulette is a casino game named for its hardware. In French the word simply means "little wheel," and English used it that way before it meant the game.
Etymonline traces it to Old French roelete, "little wheel," from the 12th century. That word was modeled on Late Latin rotella, a diminutive of Latin rota, "wheel." English first recorded "roulette" in 1734 in that plain sense of a small wheel. By 1745 it meant the gambling game with a revolving disk.
Before the French word took over, English had its own name for a similar game. Etymonline notes "roly-poly" or "rouly-pouly" from 1712.
The game's own beginnings are hazier than its name. Wikipedia reports that many historians believe Blaise Pascal introduced a primitive form of roulette in the 17th century. In that telling, he hit on it while trying to build a perpetual motion machine. Wikipedia presents this as a belief, not a documented fact.
The firmer evidence starts in the 18th century. Wikipedia describes the mechanism as a hybrid of a gaming wheel invented in 1720 and an Italian game called Biribi. The earliest reference to a game called roulette is a 1758 rule in New France, the French colony in North America. It banned "dice, hoca, faro, and roulette."
The modern game is recorded in Paris by 1796. An early description comes in Jaques Lablée's novel La Roulette, ou le Jour, published in 1801, which pictures a wheel at the Palais Royal.
The wheel kept changing after the name was fixed. Wikipedia says François and Louis Blanc introduced a single-zero wheel at Bad Homburg in 1843 to compete with other casinos. When Germany banned gambling in the 1860s, the Blanc family moved to Monte Carlo, where single-zero roulette became the main game. In the United States, a double-zero wheel spread from New Orleans and stayed dominant.
The same Latin rota turns up in other English words. Etymonline links rowel, the spinning star on a spur, to the same Old French "small wheel" as roulette.