Pai Gow (meaning "make nine") is used to describe two different Chinese gambling games. The Cantonese transliteration is sometimes spelled Pai Kow and in Mandarin it's Pai Jo.
If you are looking for information about the version of Pai Gow played with playing cards then check out m_turner's excellent writeup in Pai Gow Poker. When people in the United States say "Pai Gow" they usually mean Pai Gow Poker.
The other Pai Gow game is played instead with dominoes. It is an ancient Chinese and Korean game that uses the standard 32 tile Chinese domino set. This game is sometimes called "Domino Pai Gow" or "Pai Gow Tiles" to distinguish itself from the card game. The playing card and tile versions of the game are very similar and share many rules. If you are familar with the card version you'll find this game much easier to understand. Domino Pai Gow is by far the most complicated game I've ever witnessed in a casino.
You'll find Pai Gow offered in many casinos in Nevada, California, Michigan, Connecticut, New Jersey, Canada, Australia, and New Zealand. I haven't seen the game played in any of the casinos in the Midwest U.S. or in the Southern U.S. (presumably due to the smaller numbers of Asian players available in those areas).
This game utilizes a banker and up to seven other players (aka punters). (Similar to the game of baccarat, chemin de fer, and Pai Gow Poker). When you first come to a table you are required to play as the player. If you choose to play as the banker you must cover all the wagers of the other players. Some casinos will agree to "co-bank" with you splitting all your wins and losses. The casino dealer running the game gets a hand to play just as the other players. The dealer also takes his turn playing both player and banker. The dealer always plays his hand according to a "house way" which is written down and varies slightly at each casino. If you are banker and co-banking with the casino then you are required to play your hand the "house way".
To begin the dealer shuffles all the tiles and places them face down into stacks of 4 tiles each. All of the players must then place their wagers. Each player rolls 3 dice to determine which player gets the first stack of tiles. The remaining stacks are distributed in order to the remaining players (and banker) counter-clockwise.
Each player then must create two "hands" consisting of two tiles each. There is a high hand and a low hand and everyone must play their hands so that their high hand is higher ranked than their low hand. The actual ranking of the hands is very complicated and there's no easy way to explain how it's done. The casino will provide you with a hand ranking chart to help you out. All the high ranking hands have special names: Supreme Pair (or Gee Joon which is also is the name of a restaurant in Binions Horseshoe Casino), Heaven, Earth, Man, Goose, Flower. There are also higher ranking hands called Wongs and Gongs. Each individual domino also has a special name and rank (which is important because the highest domino rank is used to break ties when the dealers and players hands are tied.
One interesting thing about the game is that when a hand isn't one of the "special hands" the game plays very much like baccarat. The total of the spots on the tiles are added up and the closest to 9 is the highest hand. Also if the total is above 10 the first digit is removed from the total as in baccarat. (e.g. a total of 13 is ranked as a 3 and a total of 25 is ranked as a 5). This feature of trying to get closest to nine is how Pai Gow got its name.
Check out http://www.thewizardofodds.com/game/paigowtiles.html to see pictures of all the ranked hands and ranked tiles.
All the players and the banker must set their hands without looking at each others tiles. Once all the hands are set they are all turned face up. The player's high hand is compared to the dealer's high hand and then their respective low hands are compared to each other. The player wins even money on his wager if his high hand and low hand both beat the banker's high and low hands respectively. The player loses only if both his hands are lower than the bankers hands. If the player wins one hand and loses the other then his bet is a tie (or "push") and his wager is returned.
The banker pays all the winning bets and collects all the losing bets. If one of the players has taken on the role of the banker then he must pay the casino a 5% commission on his total winnings (if any).
The player to the banker's right has the option of becoming the banker for the next round if he so desires.
The House Edge:
The house edge for a person playing perfect strategy as a player is 1.6225%. The house edge for a banker playing perfect is -0.1472% (A positive expectation!). However you cannot just sit down and elect to play banker. The banker position must be taken in turns by all players so the effective combined house edge of playing both a perfect banker and player game is 0.7377%.
The game moves very slowly due to all these complications and with such a low house edge most casinos have very high limits on their Pai Gow tables (often $25 to $100 minimum tables). They need the players to wager lots of money to keep the casino's house hold on an even keel with the other table games offered.
Pai Gow "Optimal Strategy"?
The "optimal strategy" for this game (like all other casino games): DO NOT PLAY THIS GAME! Take whatever chips you have remaining and cash them in at the cashier cage. Pai Gow is not a beatable game using any strategy over the long run. See negative expectation.
If you must play the game then first I would suggest you purchase a Pai Gow strategy book and practice a while (either by using your own tiles or by using the free java applet below). There are only 3 possible ways that you will have to play your 4 tiles so there's not a whole lot of decisions to make, but the decisions are very important if you want to keep the house edge as low as possible.
In order to get a positive expectation while playing the banker you must not be playing identical to the "house way". You will need to adjust your strategy to play the best against a person that is playing the "house way". This being the case you must not ever co-bank (as stated previously if you co-bank with the dealer you are required to play the "house way"). If you can play perfect banker then your strategy is to encourage the other players to bet much larger amounts than you make against them. You want the majority of your wagers to be during this brief time when you are the banker.
A FREE Pai Gow Java Applet
The following link provides a simulation of a casino Pai Gow game. You can use this program to practice your moves and to check your moves against the optimal moves.