If more than one player at the table is still in a hand, and the final betting round has ended, the game has entered a phase known as the showdown (the part where people expose their cards to see who has won).
Anyone still in the hand may turn over his cards at this point, but people are usually reluctant to do so. If no one is making a motion to show his hand, the last person to raise (or the first person to bet if there were no raises on the river) in the final round of betting exposes his hand first — the theory being that this is the hand that everyone else called. (If no one bet on the river, the player closest to the dealer has to expose his hand first.)
Players are welcome, if not encouraged, to muck a losing hand without showing it. After a hand has been mucked (meaning turned face-down and pushed into the discards beyond the player's control), the hand is considered dead and no longer in competition for the pot.
After the first hand is exposed, remaining players expose their hands in a clockwise order. The dealer will muck losing hands one by one, leaving only the winning hand exposed for the awarding of the pot. When the dealer has determined a winner, she will push the pot in the direction of the player, and at that point, the winner is welcome to drag the pot in.
Don't ever trust another player when he announces a hand at the end of a showdown and muck your hand based on that information. After you've mucked your hand, it is truly dead and you can never bring it back. If there is an error on a player's call, other players may lie to you as a "joke." It's even more likely that they'll misread a hand. If you have any question as to who won a hand when you're involved in a showdown, merely turn your hand over and let the dealer make the decision.