If you look at the American roulette wheel and layout you will notice that there are two zeroes (green coloured). This is the main difference between the American and the European roulette games.
In comparison this means that the American roulette wheel has 38 numbers (00 and 0 to 36) while the European wheels have 37 numbers (0 to 36).
The American roulette numbers are: 0-28-9-26-30-11-7-20-32-17-5-22-34-15-3-24-36-13-1-00-27-10-25-29-12-8-19-31-18-6-21-33-16-4-23-35-14-2
The payouts for the bets in both American and European roulette are the same. For example a straight up (single number) bet pays 35 to 1 for both types of roulette games. This though makes the American roulette game to have odds more in favour of the casino. The house edge in American roulette is 5.26% while the European roulette house edge is 2.63%. This makes it better for players to play on a European roulette wheel with a single zero.
You can read more information on the specialised roulette casino site I Love Roulette - American Roulette Wheel
The roulette odds favour the player while playing European roulette or a sinlge zero wheel.