Called (or call) bets or Announced Bets
Although most often named "Call Bets" technically these bets are more accurately referred to as "announced bets". The legal distinction between a "Call Bet" and an "Announced Bet" is that a "Call Bet" is a bet called by the player without him placing any money on the table to cover the cost of the bet. In many jurisdictions (most notably the United Kingdom and Nevada USA) this is considered gambling on credit and is illegal. An "Announced Bet" is a bet called by the player for which he immediately places enough money to cover the amount of the bet on the table, prior to the outcome of the spin / hand in progress being known.
There are different number series in roulette that have special names attached to them. Most commonly these bets are known as "the French bets" and each covers a section of the wheel. For the sake of accuracy, and at risk of being called pedantic, Zero spiel although explained below is not a French bet, it is more accurately "the German bet". Players at a table may bet a set amount per series (or multiples of that amount). The series are based on the way certain numbers lie next to each other on the roulette wheel. Not all casinos offer these bets, and some may offer additional bets or variations on these.
Voisins du zero ("neighbors of zero")
This is a name, more accurately Grand Voisins du Zero, for the seventeen numbers which lie between 22 and 25 on the wheel including 22 and 25 themselves. The series is 22,18,29,7,28,12,35,3,26,0,32,15,19,4,21,2,25 (on a single zero wheel).
9 chips or multiples thereof are bet. 2 chips are placed on the 0,2,3 trio; 1 on the 4/7 split; 1 on 12/15; 1 on 18/21; 1 on 19/22; 2 on 25/26/28/29 corner; and 1 on 32/35.
The bet on this section is very popular in Eastern Europe, most notably in the Czech Republic.
Jeu zero ("zero game")
Zero game, also known as zero spiel (spiel is German for game), is the name for the numbers closest to zero. All numbers in the zero game are included in the big series, but are placed differently. The numbers are as follows: 12, 35, 3, 26, 0, 32, 15.
The bet consists of 4 chips or multiples thereof. 1 chip on 0/3 split, 1 on 12-15 split, 1 on 26 straight up and 1 on 32-35 split. Popular bet in Germany and many European casinos. This bet is also offered as a 5 piece bet in many Eastern European casinos. As a 5 piece bet it is known as zero spiel naca and includes, in addition to the chips placed as noted above, a straight-up on number 19.
Tier ("the third")
This is the name for the twelve numbers which lie on the opposite side of the wheel between 27 and 33 including 27 and 33 themselves. The series is 27,13,36,11,30,8,23,10,5,24,16,33 (on a single zero wheel). The full name (although very rarely used - most players just call it as "tier") for this bet is la tier du cylindre (translated from French into English means one third of the wheel) because it covers twelve numbers (placed as 6 splits), which is as close to 1/3rd of the wheel as one can get. Very popular bet in British casinos. Tier bets out number Voisin and Orphans bets by a massive margin.
6 chips or multiples thereof are bet. 1 chip is placed on each of the following splits: 5/8; 10/11; 13/16; 23/24; 27/30; 33/36.
The Tier bet is also called the "Small Series" and in some casinos (most notably in South Africa) "Series 5/8" It includes the following wagers which are all splits
5/8, 10/11, 13/16, 23/24, 27/30, 33/36
A variant known as "Tier 5,8,10,11" has an additional chip placed straight up on 5, 8, 10 and 11; and so is a 10-piece bet.
These numbers make up the two slices of the wheel outside the Tiers and Voisins. They contain a total of eight numbers, the Orphans comprising 17,34,6 and the Orphelins being 1,20,14,31,9.
5 chips or multiples thereof are bet. 1 chip is placed straight-up on 1 and 1 chip on each of the splits: 6/9; 14/17; 17/20 and 31/34.
"xx and the neighbors"
A number may be backed along with the 2 numbers on either side of it in a 5 piece bet. For example, "0 and the Neighbors" is a 5 piece bet with 1 piece straight-up on 3, 26, 0, 32 and 15. Neighbors bets are often put on in combinations, for example "1, 9, 14 and the neighbors" is a 15 piece bet covering 18, 22, 33, 16 with 1 piece; 9, 31, 20, 1 with 2 pieces and 14 with 3 pieces.
Any of the above bets may be combined, eg "Orphelins by 1 and Zero and the Neighbors by 1." The "...and the Neighbors." is often assumed by the croupier.
Another bet offered on the single zero game is "finals". Most often pronounced finaal, but also finale (common with Italian speakers), and finals.
Finaal 6, for example, is a 4 piece bet and consists of 1 piece placed on each of the numbers ending in 4, that is 4, 14, 24 and 34. Finaal 7 is a 3 piece bet, 1 piece each on 7, 17 and 27.
Finaal bets from finaal 0 (zero) to finaal 6 cost 4 pieces. Finaal bets 7, 8 and 9 cost 3 pieces.
Some casinos also offer split-finaal bets, for example finaal 5/8 would be a 4 piece bet, 1 piece each on the splits 5/8, 15/18 and 25/28.