There are three versions of baccarat, namely chemin de fer, baccara en banque or baccara à deux tableaux, and punto banco or baccarat. The first two date back to 19th-century France, while the third originated in 20th-century Argentina and came to Nevada in 1959; that's the version now played in the U.S. All three versions are currently played in Monte Carlo.
We assume you know the basic rules of baccarat. The house advantage for the player bet is 1.24% (pushes included) or 1.36% (pushes excluded). For the banker bet the numbers are 1.06% and 1.17%.
Chemin de fer ("railway" in French) differs from baccarat in several respects. First, it's played with a six-deck instead of an eight-deck shoe. Second, it's not a house-banked banked game and each player has the opportunity of becoming the banker as the shoe moves around the table counter-clockwise. When the banker wins, he retains the shoe and the casino collects a 5% commission. Finally, there's some flexibility in the strategy rules. Specifically, the player can draw or stand, as he wishes, with a two-card total of 5. The banker's drawing rules are identical to those of baccarat with two exceptions. If the banker has 3 and gives 9, or if he has 5 and gives 4, then he may draw or stand, as he wishes.
We emphasize that these are the current rules, but in the early 20th century, the banker's strategy was essentially unconstrained. It turns out that the optimal strategy in modern chemin de fer is exactly the mandatory strategy in baccarat (player draws to 5, banker draws in the two optional cases). Thus, the house advantage at chemin de fer is similar to that of baccarat.
Baccara en banque, like chemin de fer, is played with a six-deck shoe. It's typically banked by a syndicate that shares its profits with the casino. Instead of dealing two two-card hands, player and banker, the dealer deals three two-card hands, which may be called player 1, player 2, and banker. Bets can be placed on player 1, on player 2, or on both in equal proportions. Bets on banker aren't allowed.
The result is essentially two dependent games of chemin de fer: player 1 vs. banker and player 2 vs. banker. Player 1 announces his drw-or-stand decision, then player 2 does the same. (As in chemin de fer, only when a player has a two-card total of 5 does he have any discretion.) Only then are the third cards dealt face up.
The banker's decision as to whether to draw or stand is unconstrained. It depends on player 1's third card (if any), player 2's third card (if any), the total amount of money bet on player 1, and the total amount of money bet on player 2. A research paper on this game was published by Downton and Lockwood in the Journal of the Royal Statistical Society in 1976, and it was found that the banker's advantage is about 0.85%.
Finally, a hybrid form of chemin de fer and baccara en banque is used when there aren't enough players for a full game of the latter. Here, there's a fixed banker as in baccara en banque, but only one player hand as in chemin de fer. This game was played by British agent James Bond against Soviet agent Le Chiffre in Casino Royale (the novel, not the recent remake of the movie, which featured high-stakes poker).
We take this opportunity to correct some questionable information given in answer to the question about the origin of the term "punter" (QoD 9/12/2006).
First, baccara dates back only to the 19th century. According to Parlett (A History of Card Games, 1991), "Unsupported protestations of mythic antiquity notwithstanding, Baccara ('Baccarat' in British and Nevada casinos) does not grace the realms of recorded history before the nineteenth century, when it became firmly entrenched in French casinos." And according to Foster's Complete Hoyle (1953), baccara "originated in the south of France, and came into vogue during the latter part of the reign of Louis Philippe." (His reign lasted from 1830 to 1848.)
The earliest citation by The Oxford English Dictionary is 1866; by Trésor de la langue française is 1851; and by Grande dizionario della lingua italiana is 1855. (The earliest citation we have seen is 1847.)
Incidentally, contrary to many gambling authorities, "baccara" does not mean "zero" in Italian; indeed, the word is of French origin, albeit of unknown etymology.
There are at least two older games that are perhaps ancestral to baccara. The game of macao, sometimes called Italian-style baccara, shares card values with baccara as well as the value of the best possible hand (nine). However, modulo 10 arithmetic (e.g., 5+7= 2) isn't used. The Korean domino game kol ye-se shares with baccara the best possible hand, as well as modulo 10 arithmetic. Basset (mentioned in the "punter" answer) is a close relative of faro and is completely unrelated to baccara.
The French equivalent of "punter" is "ponte," a term used by Montmort in the early 18th century. Therefore, the conjecture that "punter" somehow derives from the game of baccara, a game that didn't even exist in the 18th century, seems unlikely.
Sadly, this brings us no closer to the actual origin of the term punter, which remains shrouded in mystery.
