Three of a kind is exactly what it sounds like: a five-card hand in which three of the cards are the same rank. In most poker games, it's a medium strength hand, and while it can be beat by a straight or a flush, it can also improve in big ways.
Some poker games involve getting three cards at once, and in these cases, you might start out with three of a kind. This is especially true in stud games, and can make it a very strong opening: opponents can only see one card, so they won't know how strong you are. (Note: in Omaha, the opposite is true: starting with a set is actually weak, because you can only use two of those cards.)
In other cases, you'll start with just one card or a pair and improve to three of a kind when more cards are dealt. In Hold'Em games, starting with a pair in the hole and then improving to three of a kind using community cards is called a set--this is distinct from holding a card that matches the rank of two cards on the board. This is because a set is more likely to improve to a full house than when you hold one card in your hand that matches two on the board: with a set, you need the board to pair just once to give you a full house.
Three of a kind is a medium-strong hand in most poker games, meaning it is likely to be strong but not impermeable. Three of a Kind is beaten by Straights and Flushes, so you might want to bet early to chase out other players who are on a draw. And don't think your three of a kind is invincible, especially if the board has paired: someone else might have filled up.
If two players have Three of a Kind, the highest set wins: so three Tens beats three Sixes. If two players have the same three of a kind, they go to the kicker: whoever has the highest card to go along with their three of a kind wins. A second kicker can break tied kickers, so if two players have three Sixes and a King, but one has a Jack and the other has a Five, the player with the Jack wins. If both players have the same five-card hand, they split the pot.