Showdown disputes include who shows first, whether a discarded hand is still live, and who may ask to see a hand. Pot division, including side pots and splits, is covered in Pots.
Robert's Rules of Poker covers cash games; the TDA rules cover tournaments. Their rules differ on who may ask to see a hand (7.17, 7.18).
7.1 A showdown follows the last betting round when two or more players remain
If two or more players still hold cards when the final betting round is over, their hands are compared and the best wins. If everyone else folds, at any point, there is no showdown: the last player left wins the pot and doesn't have to show anything. The same holds if, at the end, every player but one throws their hand away face down without showing it.
7.2 After a bet on the last round, the last player to bet or raise shows first
If anyone bet on the final round, the player who made the last bet or raise turns their hand up first. The others follow clockwise, and each may show a winning hand or give up (7.15).
Example: on the river Ana bets $40, Ben raises to $120 and Ana calls. Ben made the last aggressive action, so Ben shows first, even though Ana bet first.
7.3 When the last round is checked, the first player to act shows first
If nobody bet on the final round, the player who would have acted first on it shows first. In Hold'em, Omaha and other button games that is the first player still in to the left of the button. In stud it is the player with the best hand showing, and in Razz the lowest hand showing. Robert's Rules and the TDA agree on this; Robert's also applies it when everyone is all-in and there was no betting on the last round.
7.4 Anyone may show early, and a likely winner is asked to
Show order exists to settle who has to go first when nobody wants to. It doesn't stop anyone tabling a hand sooner. Robert's Rules encourages a player with a probable winner to show it straight away, to keep the game moving; the TDA uses its order only when nobody shows of their own accord. With side pots, Robert's asks players to wait until the pot they are in is being settled.
In a tournament, once a player is all-in and all betting is over, every hand still in is turned face up at once, before the rest of the board is dealt, and nobody in the main pot or a side pot may throw a hand away without showing it. Robert's Rules has the same rule for tournaments. Its cash-game showdown rules have no such requirement, so in a cash game the normal show order (7.2, 7.3) applies unless the house says otherwise.
At showdown a hand is worth what its cards make, whatever its owner says. Saying "flush" with a straight doesn't make it a flush, and failing to notice your own straight doesn't lose it, provided the cards are face up on the table. This rule, cards speak, is used by every card room and by both the TDA and Robert's Rules.
Some private games instead ask each player to announce their hand and hold them to it. That is a house rule; pagat.com records it, and a group playing that way should say so up front.
What you say about your hand isn't binding, but lying about it to get an opponent to throw a winner away is a different matter. Robert's Rules calls a deliberate miscall unethical and allows the pot to be forfeited; the TDA allows a penalty. Lowball is stricter: under Robert's Rules you may not claim a better hand than you hold, and a miscall that makes another player foul their hand kills yours.
Example: in deuce-to-seven lowball, Ana announces "eight low" and Ben, holding a nine low, throws his cards away. Ana actually has a pair. Ana's hand is dead.
7.8 The dealer reads the hands, and anyone may point out a mistake
The dealer reads the tabled hands and announces the winner, but players are still responsible for keeping hold of their cards until the pot is awarded. Anyone at the table, even a player not in the hand, may point out a hand that has been misread or a pot about to go to the wrong player, and Robert's Rules says they ought to.
To claim any part of a pot, a player has to turn all their cards face up on the table, where the dealer and the other players can read them: both cards in Hold'em, all four in Omaha, all seven in 7-Card Stud. That counts as a tabled hand. Showing one card that would win isn't enough; the dealer should ask for the rest, and the floor is called if the player refuses.
7.10 Playing the board still means showing your cards
When the five board cards are your best hand, you still have to show your hole cards to share the pot. Under the TDA you table them like any other hand. Robert's Rules, for cash games, says you must declare that you're playing the board before throwing your cards away, or you give up your claim; for tournaments it says keep your hand and show it if asked.
7.12 A hand thrown face down isn't dead until it's lost in the muck
Rooms differ on how final a face-down toss is. The TDA says cards thrown away face down are not automatically dead: if they can still be picked out and identified for certain, the player may table them. The dealer kills them only once they're pushed into the muck or can no longer be identified. Robert's Rules says cards thrown into the muck may be ruled dead, but lets the management retrieve a hand that is clearly identifiable if that is best for the game, especially when the player folded because they were given wrong information. Some card rooms post a stricter house rule: Mount Airy's, for example, says cards thrown face down toward the discards can't be retrieved.
Cards thrown into another player's hand are dead, face up or face down.
7.13 A winner who throws the hand away loses the pot
A player who mucks without tabling, believing they've lost or that they've won, does so at their own risk. If the cards can't be recovered and identified with certainty, the player has no claim to the pot. The TDA adds that a player who bet, then threw their hand away believing they had won while an opponent still held cards, loses any claim to a refund of called bets; only an uncalled part of a bet comes back. Hold your cards until the pot is pushed to you.
7.14 Losing hands are killed before the pot is pushed
The dealer kills the losing hands before awarding the pot. Under the TDA, a tabled losing hand is turned face down but kept apart from the muck, still identifiable, until the winner is settled and the pot awarded, in case the reading is challenged.
Once all the betting is over, a player who can see they're beaten may throw the hand away instead of showing it. Under Robert's Rules the other players still keep their right to ask to see it (7.17). If everyone but one gives up this way, that player wins without showing (7.1).
Conceding out loud is just talk. If your cards are face up and they're the best hand, they win, even if you said "you've got me". The TDA lists "you win" among the statements that don't bind anyone at showdown. What loses the hand is letting go of the cards before they're tabled (7.13).
7.17 Robert's Rules: anyone dealt in may ask to see a hand
In a cash game played by Robert's Rules, any player who was dealt into the hand may ask to see any hand that could have taken part in the showdown, even one already mucked. It is a privilege, and the house can take it away from someone who abuses it. Asking has a consequence: if a player other than the winner asks to see a hand that was thrown away, that hand is shown but stays dead. If the winner asks to see a loser's hand, both hands are live, and the better one wins.
7.18 The TDA: a river caller may insist; everything else is the director's call
In a TDA tournament the right is narrower. If there was a bet on the river, a player who called it may have the last aggressor's hand tabled, provided the caller tabled their own hand or still holds it. Every other request, such as seeing another caller's hand or any hand when the river was checked, is up to the tournament director. A player who has already mucked, or no longer holds cards, loses the right to ask at all.
Example: on the river Ana bets $40 and Ben calls. Ben tables two pair and Ana throws her cards in face down. Under both books, Ben may ask to see Ana's hand. Cal, who folded on the turn, could ask under Robert's Rules, because he was dealt in; under the TDA he no longer holds cards, so he can't.
7.19 Showing after an uncontested win is up to you
A player who wins because everyone else folded never has to show. They may choose to show some or all of their cards once the pot is theirs, but what one player is shown, the whole table may see (9.7).
7.20 A misread hand can be challenged until the next hand starts
A mistake in reading a tabled hand can be raised until the next hand begins. Robert's Rules marks that point as the first riffle of the next shuffle (6.2). The TDA allows a mistake in counting or paying the pot to be raised a little longer, until substantial action on the next hand (6.15), and, for a hand that ends at a break, one minute after the pot is awarded. Fixing a pot that was paid to the wrong player is 8.19.
7.21 Ties and odd chips are settled by the pot rules
When two or more hands tie, the pot is split, and any chip that won't divide evenly goes to a player chosen by position or by suit, depending on the game. Those rules are 8.7 to 8.10.