Royal Flush: A royal flush is a hand where you have five cards 10-J-Q-K-A that all happen to be from the same suit. This is the highest ranking hand in non-wildcard poker, regardless of the variant.
Straight Flush: A straight flush is a hand where you have five cards in sequential order (i.e. 3-4-5-6-7) that also happen to have the same suit (i.e. hearts). This is the second highest poker hand by the non-wildcard rankings, regardless of the variant that is being played.
Four of a Kind: Each ranked card has four appearances in a standard deck of card; one of each of the four suits (diamonds, clubs, hearts and spades). This means that there are four aces, four kings, four queens and so on. When you have all of the four cards of that particular rank, it is known as having four of a kind. Four of a kind is also sometimes referred to as quads.
Full House: When you have a full house, you have three of one particular rank and two of another particular rank (i.e. three fives and two aces would make a full house). The full house in the bracketed example can also sometimes be called fives full or fives full of aces.
Flush: When you have a flush, it means you have five cards of the same suit (i.e. five spades or five diamonds).
Straight: When you have a straight, it means that you have five cards that are in sequential order. The difference between a straight and a straight flush is that with a straight the cards do not also have to be of the same suit; they just have to be in sequential order.
Three of a Kind: When you have three of a kind, it means that you have three cards of the same rank (i.e. three fives). Three of a kind is also sometimes referred to as a set or as trips.
Two Pair: When you have two pair, it means that you have two sets of two cards with the same rank (i.e. three fives and three aces). In the bracketed example, the higher ranked pair is aces and therefore this two pair might also be called Aces Up.
High Card: If you have none of the other poker hands, then your hand value is determined by the highest card in your hand. For example, if your hand was A-J-6-5-2, then your official hand title would be Ace High or in the case of someone else also having an ace, Ace-Jack High.