In Texas Hold em, each player gets two face down cards (the Hole) and everyone bets. Then three shared Community cards are dealt face up (the Flop). There is a second round of betting. Another Community card is dealt (the Turn), followed by another round of betting. Then the last Community card (the River) is dealt and players make their final bets.
Each player remaining makes the best possible five-card poker hand (based on traditional Poker Hand Rankings) using any combination of their two Hole cards and the five Community cards. The player with the best hand takes all the chips (or money) in the pot. If the best possible hand consists of exclusively Community cards, the remaining players (those who haven't folded out of the hand) split the pot.
In Texas Holdem, your position at the table relative to the dealer can have definite advantages and disadvantages. To be fair to all players, since casinos (online and off) provide their own dealer, the dealer's position at the table rotates. To begin each new hand, the "Dealer Button" (or just "the Button"), the disc marking the position of the dealer relative to the players at the table for that hand, is moved one player to the left.
In Texas Holdem, in place of antes two players each hand post "Blinds", forced bets that ensure action (stakes) in the game. The player to the left of the Button posts the "Small Blind", usually half the minimum bet. The player to their left posts the "Big Blind", usually equal to the minimum bet. (ie. In a fixed limit, $5/$10 game, the small blind is $2.50 and the big blind is $5). In heads-up (one-on-one) hands, the player who posted the small blind is also considered "on the button".
When a player first joins a Texas Holdem poker table (other than in a tournament) he/she has the option of immediately posting the big blind at the beginning of the next hand and being dealt-in right away, or waiting until the big blind naturally comes around to their position and joining the game then.
Each player is dealt two Hole cards face down. A player's Hole cards are the only cards that will distinguish their hand from his/her opponents'.
First Round of Betting
The player to the left of the big blind starts; either checking, betting (at minimum the big blind), or folding. Once a bet is placed, each subsequent player in turn can call that bet, raise it, or fold. When the bet comes around to the player in the small blind that player, if they wish to call, only needs to post the difference between the current bet and the small blind. When the bet comes around to the big blind, that player, to stay in, must post the difference between the current bet and the big blind. If the only bet made was the minimum bet (calling the big blind) the player with the big blind does not need to add any chips to the pot and can simply "check".
Three Community cards are dealt face up. Many players, though certainly not all, will only stay in at this point if the Flop improves their hand.
Second Round of Betting
This round and all subsequent rounds of betting now begin with the player to the left of the button (the player who posted the small blind). This put the player "on the button" at an advantage in that they are last to bet for the rest of the game and can therefore gather valuable information on their opponents before deciding what to do.
The fourth face up Community card is placed on the board.
Third Round of Betting
Usually the minimum bet doubles and remains at this level for the next (and final) betting round as well.
The last Community card is laid face up on the board. Each player now knows exactly what hand he/she is holding.
Fourth & Final Round of Betting
Last chance to fold or bluff or siphon off more of your opponent's chips.
All remaining players show their hands. The winner, the player with the highest ranked hand, takes the pot. If there is a tie, players split the pot. If there is/are side pot(s) (see below), they are awarded accordingly. A player may only win pots that they have called completely.
In any round of betting, if a player wishes to stay in but does not have enough chips to match the current bet, a "Side Pot" is started containing all the bets above and beyond that which the player in question can call. At the end of the hand, if that player has the best hand, they win the initial pot. Then the player with the second best hand gets the side pot (provided that player was able to call all bets). If a player other than the one who bet all their chips is the winner, that player gets both pots and the player with no chips is either eliminated (in a freeze-out tournament) or must buy-in again with more chips if they want to keep playing.
Look for general Poker Strategies and specific Texas Holdem Strategy on this site for enhanced and improved game play. Additionally, you can find articles here on specific Limit Texas Holdem and No Limit Texas Holdem rules.