Two pair is five-card hand in which two sets of two cards are the same rank: a pair of one card plus a pair of another. The fifth card is called the kicker.
Most poker hands don't start by giving players four cards, so most two pair are made in later rounds of betting. The only exception is Omaha: because you must use exactly two hole cards, a player holding two pair in their hand does not actually have two pair and must use the board to improve.
Two pair is a medium-strong hand, and if you feel you have the best hand, you can usually convince other players to build the pot. You can bluff, slowplay or value bet to accomplish this. Plus, there's always the chance of filling up and turning your two pair into a full house.
If two players have two pair, the player with the higher pair wins the hand--so a hand of two Jacks and two Tens beats a hand of two Tens and two Nines. If the higher pair is the same, then the lower pair breaks the tie: two Aces and two Fives beats two Aces and two Threes. If they have the same two pair, the higher kicker breaks the tie, and if they have the same five-card hand, they split the pot.
In Hold'Em games, a player's two pair can be counterfeited. This is when a player has two pair using lower cards on the board--for example, the player holds a Six and a Seven, and the board shows Six, Seven, and Jack on the flop, so the player has two pair, Sevens and Sixes. But if the turn comes up with another Jack, now the player has two pair, Jacks and Sevens--the two Sixes don't mean anything. Furthermore, if another player in the pot has a Seven in their hand--especially a Seven with a high kicker--they can win the pot, even though the first player had the stronger hand, to begin with. This is called counterfeiting because it weakens the first player's hand. If you ever look down and think you have "three pair," think again: you might have had your hand counterfeited.