Like all poker games, Hold'em has a very specific order in which the cards are dealt and played.
Hole cards
At the start of a Hold'em hand, after the two blinds have been posted, all players are dealt two cards facedown. These are known as the hole or pocket cards. Players then make a decision to call the blinds (match the big blind), raise the blinds (increase the bet) or fold (quit playing and throw their cards facedown to the middle of the table).
In the form of Hold'em known as Limit, the bets have to be of a certain specified amount. In No-Limit, players may bet any amount of their chips on the table.
If you're a newbie to the game, you might consider this: If both of your hole cards are not 10s or greater (Jacks, Queens, Kings, or Aces), fold. Yes, it sounds harsh, but it'll keep you pretty much only playing the cards that you should — and about the right frequency of hands.
Make sure not to show your hole cards to other players at the table (even if those other players are no longer in the hand). And after you've looked at your cards, you should protect them from being collected by the dealer by placing an extra poker chip (or some other small object) on top of them.
The flop
After the betting action is done on the round with the hole cards (also known as pre-flop), three cards are displayed by the dealer simultaneously to the center of the poker table — this is known as the flop. At this point, each player at the table has a unique five-card poker hand consisting of his two hole cards and the three community cards.
Because of the raw number of cards involved, the flop typically gives you the general tenor of the poker hand and definitely gives you a good idea of the kind of hand to look for as a winner. For example, an all-Spade flop (especially with a lot of players still in the hand) will be hinting at a flush as a strong possibility for a winner.
Betting begins with the first person still in the hand to the left (clockwise) of the dealer button. As a general rule, you want your hand to match the flop, and you should fold if it doesn't.
In Limit play, the size of the bet you can make on the flop is identical to the amount you can make pre-flop.
The turn
After the flop betting round is completed, another community card is placed, known as the turn (or sometimes fourth street). Each of the remaining players now has a six-card poker hand made up of his two private hole cards and the four community cards. Hold'em is a game where only five cards count toward a poker hand, so everyone has a theoretical "extra" card at this point.
In Limit, the betting is now twice the amount that was bet pre- and post-flop.
Poker wags like to say, "The turn plays itself," meaning your hand gets better and you bet it, or it doesn't and you start giving strong thoughts to folding. This is more or less true.
The river
After the betting round of the turn, a final community card is exposed, known as the river (sometimes called fifth street).
Each player left in the game has his final hand consisting of the best five cards of the seven available (two private hole cards and the five community cards). Players may use two hole cards along with three community cards, one hole card combined with four community cards, or just the five community cards (known as playing the board). Again, poker hands are made up of the best five cards — the other two available to any given player don't count. There is one final round of betting.
The showdown
The showdown is what happens after the final river bets have been placed. Although it isn't formally required, typically the person who initiated the final round of betting is first to show her hand. The action then proceeds in a clockwise fashion with players either mucking their hands if they can't beat the hand exposed, or showing a better hand (at which point the dealer mucks the old, "worse" hand and continues around the table for any remaining hands).
Winners and losers are determined by the standard poker hand rankings.
If you're ever unclear about who is winning a hand, just turn your cards face up and let the dealer decide. Never take a player's word on what she has in hand until you've actually seen her cards with your own eyes — when you muck a hand, it's officially dead.