Each person in the game is dealt one card, followed by a second card. The cards are dealt clockwise, starting with the person to the left of the dealer. It is important that these "hole" cards are deal face down so that the other players in the game can not see them. If you're playing online, this will automatically happen correctly. If you're playing Texas Hold'em at a casino or with a group of friends, there will be rules regarding any mis-deals.
The person to the left of the dealer will put in the "small blind" and the player to the left of them will put in the "big blind". The next person to the left will start the betting round. When there are only two players left, special rules apply for the small and big blinds: the small blind is posted by the dealer and the big blind by the other remaining player.
Once all the betting has finished for the pre-flop, the dealer "burns" one card by dealing it face down and then deals the next three community cards face up on the table. Each player can now use these cards as well as their hole cards to create their best hands at this stage of the game and bet respectively.
Again, the dealer burns one card and then deals a fourth community card. As with the flop, players can use this fourth card to make up their best hand. There will be a round of betting if, if the turn has been disappointing, a round of people checking!
This is the final community card to be deal, again after one card has been burned. This is the last opportunity for betting for this hand.
Assuming that there is more than one player left in the hand, all remaining players show their cards. This happens clockwise from the dealer.
Each remaining player in the hand uses the best five cards from a combination of their own two hole cards and the five community cards. If two or more players have the same best hand at this stage of the game, the pot will be split between them.
At any stage of a game, a player is allowed to "fold" their cards and leave the hand. Obviously if they do this, they will not be able to win the hand and any bets that have been made during the hand will be lost.