General Rule: When hands tie on the rank of a pair, three of a kind, etc, the cards outside break ties following the High Card rules.
An Ace high straight-flush is called a Royal Flush and is the highest natural hand.
A straight flush is the best natural hand. A straight flush is a straight (5 cards in order) that are all of the same suit. An ace may either be high low (5-4-3-2-1) but a straight may not 'wraparound' such as e.g. K-A-2-3-4.
Four of a kind
Four of a kind is simply four cards of the same rank.
A full house is a three of a kind and a pair. Ties are broken first by the three of a kind, then pair. So K-K-K-2-2 beats Q-Q-Q-A-A, which beats Q-Q-Q-J-J.
A flush is a hand where all of the cards are the same suit, such as J-8-5-3-2, all of spades. When flushes ties, follow the rules for High Card.
A straight is 5 cards in order, such as 4-5-6-7-8. An ace may either be high (A-K-Q-J-T) or low (5-4-3-2-1), but the straight may not 'wraparound'. When straights tie, the highest straight wins. If two straights have the same value (AKQJT vs AKQJT) they split the pot.
Three of a kind
Three cards of any rank, matched with two cards that are not a pair. Again, highest three of a kind wins.. If both are the same rank, then the compare High Cards.
This is two distinct pairs of card. The highest pair wins ties. If both hands have the same high pair, the second pair wins. If both hands have the same pairs, the high card wins.
One pair with three distinct cards. The highest pair wins and the high card breaks ties.
This is any hand which doesn't qualify as any one of the above hands. If multiple people tie for the highest card, they look at the second highest, then the third highest etc. High card is also used to break ties when the high hands both have the same type of hand (pair, flush, straight, etc).