Any poker hand consists of five cards- not six, nor seven, not four nor three - but five. The ranks of the poker hand, from highest to lowest, are listed further down. Any hand in a higher category beats any hand in a lower category (so for example any three of a kind beats any two pairs). Between hands in the same category the rank of the individual cards decides which is better, as described in more detail below.

Maybe people might wonder why even mention the terrible case of (say) two threes of a kind of equal rank. This obviously cannot arise in standard poker, but such comparisons are needed in some other games using poker combinations - for example poker menteur - and in games with wild cards.

1. Royal Flush
Oh, dear, this is the highest poker hand. It consists of ace, king, queen, jack, ten, all in the same suit. As all suits are equal, all royal flushes are equal.

2. Straight Flush
Five cards of the same suit in sequence - such as J-10-9-8-7. Between two straight flushes, the one containing the higher top card is higher. An ace can be counted as low, so 5-4-3-2-A is an on going straight flush, but its top card is the five, not the ace, so it is the lowest type of straight flush. The cards cannot "turn the corner": 4-3-2-A-K is not valid.

3. Four of a kind
Four cards of the same rank - such as four queens. The fifth card can be anything- but never can it be one of the four. This combination is sometimes known as "quads", and in some parts of Europe it is called a "poker", though this term for it is unknown in English. Between two fours of a kind, the one with the higher set of four cards is higher - so 3-3-3-3-A is beaten by 4-4-4-4-2. It can't happen in standard poker operation, but if in some other game you need to compare two fours of a kind where the sets of four cards are of the same rank, then the one with the higher fifth card is better.

4. full house rank
This consists of three cards of one rank and two cards of another rank or the other way around- for example three sevens and two tens (colloquially known as "sevens full" or more specifically "sevens on tens"). When comparing full house ranks, the rank of the three cards determines which is higher. For example 9-9-9-4-4 beats 8-8-8-A-A. If the threes of a kind were equal, the rank of the pairs would decide.

5. Flush
Five cards of the same suit. When someone cares to go on comparing two flushes, the highest card determines which is higher. If the highest cards are equal then the second highest card is compared; if those are equal too, then the third highest card, and so on. For example K-J-9-3-2 beats K-J-7-6-5 because the nine beats the seven.

6. Straight
Five cards of mixed suits in sequence - for instance Q-J-10-9-8. When comparing two sequences, the one with the higher ranking top card is better. Ace can count high or low in a straight, but not both at once, so A-K-Q-J-10 and 5-4-3-2-A are valid straights, but 2-A-K-Q-J is not. 5-4-3-2-A is the lowest kind of straight, the top card being the five.

7. Three of a Kind
Three cards of the same rank plus two other cards (any cards). This pretty good combination is also known as Triplets or as some call it Trips. When comparing two threes of a kind the hand in which the three equal cards are of higher rank is better. So for example 5-5-5-3-2 beats 4-4-4-K-Q. If you have to compare two threes of a kind where the sets of three are of equal rank, then the higher of the two remaining cards in each hand are compared, and if those are equal, the lower odd card is compared.

8. Two Pairs
A pair is two cards of equal rank. Double that and you've got two pairs. In a hand with two pairs, the two pairs are of different ranks (otherwise you would have four of a kind), and there is an odd card to make the hand up to five cards. When comparing hands with two pairs, the hand with the highest pair wins, irrespective of the rank of the other cards - so J-J-2-2-4 beats 10-10-9-9-8 because the jacks beat the tens. If the higher pairs are equal, the lower pairs are compared, so that for example 8-8-6-6-3 beats 8-8-5-5-K. Finally, if both pairs are the same, the odd cards are compared, so Q-Q-5-5-8 beats Q-Q-5-5-4.

9. Pair
A poor hand with two cards of equal rank and three other cards which do not match these or each other. When comparing two such hands, the hand with the higher pair is better - so for example 6-6-4-3-2 beats 5-5-A-K-Q. If the pairs are equal, compare the highest ranking odd cards from each hand; if these are equal compare the second highest odd card, and if these are equal too compare the lowest odd cards. So J-J-A-9-3 beats J-J-A-8-7 because the 9 beats the 8.

10. High Card
Five cards which do not form any of the combinations listed above- none. When comparing two such hands, the one with the better highest card wins. If the highest cards are equal the second cards are compared; if they are equal too the third cards are compared, and so on. So A-J-9-5-3 beats A-10-9-6-4 because the jack beats the ten.

Notes on Low Poker
When playing games in which the lowest hand wins, there are some modifications to the ranking. These may not be universal, so should be discussed in advance when starting a game with new players. As far as I can tell, the most usual rules are:

straights and flushes do not count as combinations
aces count as low, below the twos
a hand is always considered to belong to the highest category into which it fits - for example 7-7-7-5-5 counts as a full house - not (for example) as a pair with three odd cards that happen to be equal.
With these rules the best low hand is 5-4-3-2-A, which does not count as a straight in this case.
Notice that because aces are low, a pair of aces is the lowest, and in this context therefore the best pair, beating a pair of deuces.

Variations that I know of include:

recognising straights and flushes: in this version the best hand is 6-4-3-2-A of mixed suits;
playing with aces always high: in this version (with straights and flushes also counting) the best low hand is 7-5-4-3-2 of mixed suits.
Ranking of suits
In standard poker there is no ranking of suits. If two hands are identical apart from the suits of the cards then they count as equal. In standard poker, if there are two highest equal hands in a showdown, the pot is split between them.

However, Franco Pratesi has informed me that in Italy there is a generally accepted ranking of suits in poker. The ranking is: hearts (high), diamonds, clubs, spades (low). These ranks are used to break ties between otherwise equal hands. Other players in different places use other suit rankings - for example Ken Pikus (kpikus@chemonics.com) and John (jqv77@hotmail.com) report that their circles play with the ranking spades (high), hearts, clubs, diamonds (low); Rudolf Lercher (WLNLER@rlb-noe.raiffeisen.at) and his friends play with the ranking hearts (high), diamonds, spades, clubs (low).

Note that if you do play with a suit ranking, it is not at all obvious how it should apply when comparing hands with mixed suits, and careful discussion in advance is needed to avoid unpleasant arguments later. For example, using the Italian ranking, which of the following would you expect to be higher:

Hand A: 8-8-J-9-3 or Hand B: 8-8-J-9-3 ?
Apparently in Italy the majority view would be that hand B is higher because the J beats the J - when two hands are equal in rank, the comparison is between the highest card which is not part of the combination. A surprising consequence of this rule is that K-Q-7-6-2 beats K-Q-7-6-2. The combination is "high card", so the kings are ignored. You compare the highest card that is not in the combination, and the higher queen wins!

Poker Hand Ranking with Wild Cards
A wild card is a particular card, often a joker, which can be used to substitute for any card the holder wishes, even a duplicate of a card the holder already has. Several cards may be designated as wild - for example all the twos. This must be agreed in advance.

The hand ranking is the same as described above, except that it is now possible to have five of a kind - five cards of the same rank - in which of course at least one will be represented by a wild card. Five of a kind is the highest combination, beating a Royal Flush.