Poker hand rankings

1
Royal flush
Ace, king, queen, jack and ten, all of one suit. The best hand there is.
2
Straight flush
Five cards in a row, all of one suit. The higher top card wins.
3
Four of a kind
Four cards of the same rank. Also called quads.
4
Full house
Three of a kind and a pair. The three decide first: jacks full of eights beats tens full of aces.
5
Flush
Any five cards of one suit. The highest card decides, then the next.
6
Straight
Five cards in a row, any suits. The ace can be high (A-K-Q-J-10) or low (5-4-3-2-A).
7
Three of a kind
Three cards of the same rank. Called a set with a pair in your hand, trips with a pair on the board.
8
Two pair
Two different pairs. The higher pair decides, then the lower, then the fifth card.
9
One pair
Two cards of the same rank. The other three cards, the kickers, break ties.
10
High card
No pair or better: the hand is named after its highest card.

Greyed cards don't make the hand; they're the kickers, which only count to break a tie.

Ties and kickers

A hand is always exactly five cards. When two players have the same kind of hand, compare the cards that make it, highest first, then the kickers.

A pair of aces with a king beats a pair of aces with a queen: the king kicker wins.
Suits never break a tie. Two flushes of the same cards in different suits split the pot.
The board can play. If the five cards on the board are better than anything you can make with yours, everyone still in shares that hand, and the pot is split.

When a pot is split and the chips don't divide evenly, the odd chip goes to the winner nearest the left of the button.

A flush beats a straight, because flushes are rarer.
A full house beats a flush.
A-2-3-4-5 is a straight (the wheel), the lowest one. But Q-K-A-2-3 is not: straights don't wrap round.
In Omaha, you must use exactly two of your four cards. Four hearts on the board and one in your hand is not a flush. More on Omaha.