A kicker is the set of cards in a standard five-card poker hand which are not the part of the hand which makes its rank. The kicker is used to break ties between poker hands of otherwise equivalent rank.
For example, in a poker hand of "one pair", the kicker is all three cards in the hand that are not part of the pair. In a poker hand of "two pair", the kicker is the lone remaining card that is not part of the two pair.
Poker hands which require all five cards to "make their hand" (this includes straights, flushes, full houses, and straight flushes) contain no kicker.
In Hold 'em, the term kicker may also apply to the individual card in the player's two downcards which "doesn't make a hand" when their other downcard does so (that is, which isn't part of the pair of three-of-a-kind which the player has made with their other card). For example, if a player holds K-8 in their downcards and the board shows K 9 6 4 2, then the player will often refer to their 8 in their hand as their "kicker", in addition to referring to the full set of three unpaired cards (9 8 6) which make the kicker of their full five-card hand. This is why you will often hear players say they had a better kicker than another player, or that their kicker plays.
Kickers are only mentioned or used in poker in order to break ties between hands of otherwise equivalent rank. If the kickers are also equivalent between two hands, then the hands remain tied and (in most poker variants), the pot is split among all equivalent winning hands.
Even if more cards might be available for players to use, to make theoretically larger/longer kickers (e.g. a sixth or seventh card is available for use), this is not done in standard poker hand ranking - kickers only fill out hands until the hands are five cards long.
Kickers are ranked according to the rules for ranking a hand of high card. The kicker with the highest card is the highest kicker. If two kickers have the same high card (or a high card of equivalent rank, e.g. both kickers contain a king but no ace), then the second highest card of the kicker is used to break the tie. If the second highest card is also of the same rank, then the third-highest card is used, and so on.
In this example, two players hold hands containing a pair of sevens. However, the first player has a kicker of King - Queen - Five, while the second player has a kicker of King - Jack - Nine. The first player wins, since the KQ5 kicker beats the KJ9 kicker. Kicker is used when two players are tie and does not affect the community card.
In this next example, two players in hold 'em both have four-of-a-kind, but Player 2 holds a King, and can make four sevens with a king kicker, while Player 1's best hand is four sevens with a ten kicker. Player 2 wins.