Roulette table types
First let's get an idea of the roulette table variations. The differences between the different types of tables are not huge, so I will avoid confusing you with details.
European roulette table
The European roulette table has only one zero (0) and offers better payouts than the American roulette, which has two zeroes (0, 00).

European table layout   
American roulette table

American table layout 
As you can see the American roulette table is like the European with the main difference being the additional double zero (00). Although American roulette has one extra number (double zero), the payouts are the same as in the European version. This means that the player who plays American has a bigger disadvantage (5,4%) versus the European game (2,7%) because he plays against more numbers/possibilities and his probability of win is a bit lower.
French roulette table

French roulette table layout 
In the image above you can see that the French roulette table has a different layout for the bets. Maybe the biggest difference is that the place for the dozen bets is the lower left and right parts of the table (see: P12, M12, D12). P12 is the bet for the first dozen numbers (112). M12 is the bet for the second dozen numbers (1324) and D12 is the bet for the third dozen numbers (2536).
Payout wise the French roulette is the same as the European roulette, since it only has one zero and 37 numbers in total. In some casinos French roulette can be
even more advantageous for the player than the European type. Some French roulette games offer one of the two following rules, which partially protect the player when "zero" (0) comes:
 La Partage rule: The player loses only half his bet when he has bet Red, Black, High, Low, Odd or Even (Even Chance Bets) and zero (0) has come.
 En prison rule: The player bet stays intact when he has bet Red, Black, High, Low, Odd or Even (Even Chance Bets) and zero (0) has come. The outcome of the next spin will decide if the player wins or loses the bet.
Due to these rules, some French roulette games offer only 1,4% disadvantage when the players bets on Red, Black, High, Low, Odd or Even; these bets are also called Even or Simple Chance betsbecause the pay out as much as you have bet and have nearly 50% chance of winning.
Best and Worse Roulette Bets
The truth is that in general almost all bets in the roulette table are equal in terms of
Average Expectation or simply put Risk/Reward ratio. Keep in mind however these small differentiations:
 Single zero roulette is less disadvantageous for the player than European roulette: 2,7% versus 5,4%
 French roulette with special rules can be better than European, if the player bets Red, Black, High, Low, Odd or Even: 1,4% versus 2,7%.
There is however one worst wager one can make in roulette. This bet is exclusive to American roulette and is called "
Top line bet". That is betting on
0, 00, 1, 2, 3. For this wager the player disadvantage in American roulette explodes from 5,6% to an unplayable 7,9%. This is because the casino pays you 6 times your money instead of 6,2 times your money, which is already bad enough with 5,6% disadvantage. Basically you wager 5 numbers for the price of six.
Bets and payouts
 Single number.
Probability 1/37 (European or French) or 1/38 (American). Pays 36 times your money.  Split / 2 numbers / any two adjacent numbers in the table layout, like 56 or 58.
Probability 2/37 (European or French) or 2/38 (American). Pays 18 times your money.  Street / 3 numbers any 3 numbers on the same horizontal line in the table, like 10 1112.
Probability 3/37 (European or French) or 3/38 (American). Pays 12 times your money.  Corner or Carre / 4 numbers, any 4 adjacent numbers in the table layout, like 1  2  4  5.
Probability 4/37 (European or French) or 4/38 (American). Pays 9 times your money.  Doublestreet / 6 numbers / any 6 numbers in 2 adjacent streets, like 10  11  12  13 14 15.
Probability 6/37 (European or French) or 6/38 (American). Pays 6 times your money.  Dozen / 12 numbers / any of the 3 dozens, 112 or 1324 or 2536.
Probability 12/37 (European or French) or 12/38 (American). Pays 3 times your money.  Column / 12 numbers/ any of the 3 columns in the roulette table layout.
Probability 12/37 (European or French) or 12/38 (American). Pays 3 times your money.  Low / 18 numbers / Numbers: 123456789101112131415161718.
Probability 18/37 (European or French*) or 18/38 (American). Pays 2 times your money.  High / 18 numbers Numbers: 192021222324252627282930313233343536. Probability 18/37 (European or French*) or 18/38 (American). Pays 2 times your money.
 Red / 18 numbers / all the numbers with Red color: 1357912141618192123252730323436.
Probability 18/37 (European or French*) or 18/38 (American). Pays 2 times your money.  Black / 18 numbers / all the Black numbers: 24681011131517202224262829313335.
Probability 18/37 (European or French*) or 18/38 (American). Pays 2 times your money.  Odd / 18 numbers / all the Odd numbers: 1357911131517192123252729313335.
Probability 18/37 (European or French*) or 18/38 (American). Pays 2 times your money.  Even numbers / 18 numbers / all the even numbers: 24681012141618202224262830323436.
Probability 18/37 (European or French*) or 18/38 (American). Pays 2 times your money.
*With La Partage or En Prison rule, French roulette offers slightly better odds. 
Roulette bets placement 