Poker hand rankings of five-card hands, from best to worst: royal flush (A-K-Q-J-10, same suit), straight flush (five cards of the same suit, in order, e.g. J-10-9-8-7 of hearts), four of a kind, full house (any three of a kind, plus any pair, e.g. 9-9-9-3-3), flush (any five cards, same suit), straight (any five cards in order, e.g. A-2-3-4-5), three of a kind, two pairs, one pair.
Every player is dealt two cards down. These are your "pocket" cards; they belong to you only. You look at the cards and decide whether or not you want to call or raise the "blind," a predetermined forced bet by one player (the blinds go up as the tournament goes along to promote more aggressive play).
If you decide to call the blind, call another player's raise or raise yourself, you do so. In no-limit hold'em, you can raise the pot with as many chips as you have. You still have seen only two cards.
Note: You keep your two pocket cards facedown throughout the game, unless you and another player at any point have "gone all-in." This means one player has pushed all his chips in the middle, and the other player also goes all-in. Your cards are then turned faceup because there will be no more bets placed on that hand.
After everyone has a chance to enter the pot or decide not to play that hand ("fold"), three cards are turned faceup in the middle of the table. This is called "the flop." These cards and the two that will follow are community cards; they belong to everyone at the table. You now have five cards in your hand: your two pocket cards and the three cards on the flop. A round of betting ensues.
Then a fourth card is turned over. This is called fourth street, or "the turn." You now have six cards in your hand: two pocket cards and the four community cards shared by all, sitting faceup on the table.
After another round of bets, the fifth and last community card is turned up. This is fifth street, or "the river." Unless all but one fold (you can fold at any point), there is a final round of bets. After all bets are complete, everyone shows his cards.
You now have seven cards: two pocket cards and the five community cards shared by everyone. The best five cards make up your final hand, the one that wins, loses or draws. You do not have to use any of your pocket cards. If players tie for the best hand, they split the pot. This is called a "chopped pot."