Roulette is played on a bowl-shaped wheel that spins on a center axis.
A ball is rolled in the opposite direction that the wheel spins, and as the wheel slows the ball drops, eventually, into one of the slots. There are either 37 or 38 slots. The numbered slots are either red or black, and the 0 is green. On American wheels there is also a 00, also green.
Betting is still allowed even when the dealer, called a croupier, spins the wheel n and rolls the ball around the edge of the wheel, but only until the ball starts to drop out of orbit, usually twice around the bowl. If the ball falls into a slot that coincides with your bet (a number, a color, etc) then you win.
The roulette table has a different bet slots to put your chips, such as Red or Black, Odd or Even, etc. While most casino games use chips that are universally colored by denomination, with roulette each player gets his or her own color of chips. This is because players can put their chips in the same bet slot (say two people want to bet Red, for example) and the croupier needs to tell who bet what for the pay off.
Because of the unique nature of the chips you must cash them out before leaving the table.
Once all bets are placed the ball drops, the croupier puts a marker on top of the winning number on the table, all losing bets are swept off the table and then the winning bets are paid off.