Roulette is a wheel game, like the Wheel of Fortune on tv. The wheel sits in a bowl and spins on an axis. There are 37 or 38 slotted numbers on the wheel, depending on whether you're playing on an American roulette wheel or a European roulette wheel. The numbers (except for 0 or 00) are either red or black and are numbered from 1 to 36. The 0 or 00 are green.
The dealer at a roulette table is called a croupier, and he spins the wheel in one direction and rolls the ball around the edge of the wheel in the opposite direction. When the ball's spin starts to decay, no more bets are allowed, and the ball eventually falls into one of the numbered slots. If it falls into a slot that matches your bet, you win. The different bets which are available and their payouts are covered in our section on roulette betting, but note that almost all of the bets (except one) offer the same house edge.
The roulette table has a number of bets available, most of which are pretty self-explanatory, like red or black or odd or even. When you first sit down at a roulette table, you'll need to buy chips from the croupier, and roulette chips are almost always different from all other chips in the casino. Each player is assigned her own color of chips so that the croupier knows who to pay off at the end of each spin.
A particular color of a chip might have a different value at any point during the day, depending on the player's buy-in. Roulette chips cannot be cashed in at the cashier cage for this reason. Always remember to cash out your roulette chips before leaving the table; if you don't do this, you risk only getting paid the minimum amount for the chips, whether or not that's what you paid for. The croupier has no way of remembering each specific player's denomination once they've left the table and stopped playing, so always cash out when leaving the table.
A player is allowed to bet until the croupier announces, "No more bets." Any bets after that point are called "past posting", which is illegal. Past posting is the practice of placing a bet at the roulette table after you know what the result of the spin is. This is cheating, and it's not recommended, since in most states, cheating at a casino game is seriously illegal.
After the result of the spin is determined, the croupier puts a marker on top of the winning number on the table. If there are bets on that number, the marker is placed on top of the chips. Then all losing bets are swept off the table before the winning bets are paid off. Outside bets are paid off first, then inside bets. Do not place your next bet until the dealer has finished paying off all of the previous bets.