Royal Flush:
Ace, king, queen, jack, ten, all of the same suit.
Straight Flush:
A straight where all cards are of the same suit, Royal Flush is just the highest variant of a straight flush.
Four of a kind
Four cards of the same rank, ie 4 fours. 4 fours beat 4 threes, 4 queens are beaten by 4 kings.
Full House
Three cards of one rank and two cards of another, ie. three 2's and two 9's. The rank of the three cards determine strength so three 7's and two 2's beat three 6's and two aces.
Flush
Five cards suited, highest ranked card breaks ties. ie. Ace, 4, 3, 2 of clubs beats King, Jack, ten, nine and seven of clubs.
Straight
Five cards of mixed suits in sequence, highest card in sequence used to break ties. Three of a Kind
Three cards of the same rank plus two other cards, rank is used to determine winner, ie. three 7's beat three 6's.
Two Pairs:
Exactly as stated, ie. two 9's and two 3's. Ties are determined by highest rank, ie. two aces and two 2's beats two kings and two queens. If both pairs are ranked the same then the remaining card, the kicker, is used to break the tie or determines that the hands are tied.
A pair:
Two cards in a hand of the same rank. Remaining cards are used to break ties, ie. two players with two 2's then compare their highest remaining card to break the tie, if those are the same then they compare their next highest card. If all three remaining cards match then the hands are tied.
Any hands that don't yield any of the above are ranked highest to lowest with the highest card winning.