A Kicker is the high card used to determine the winner in cases when two or more players have equally strong winning hands. 

A Kicker matters only for the hands made up of less than 5 main cards (that is, the strongest hand doesn't involve all of the five cards):
  • Four of a Kind (4 cards used, 1 Kicker left)
  • Three of a Kind (3 cards use, 2 Kickers left)
  • Two Pair (4 cards used, 1 Kicker left)
  • One Pair (2 card used, 3 Kickers left)
  • High Card (1 card used, 4 Kickers left; technically, one may consider this hand to have no "cards used", but simply 5 Kickers)
The following hands involve no Kickers:
  • Royal Flush
  • Straight Flush
  • Full House
  • Flush
  • Straight
The principle behind the Kicker is pretty simple, really.
Each hand consists of 5 cards exactly, no more and no less.
If the "determining base" of the hand involves less than 5 cards, the remaining ones may help swing the game in one or the other player's favor, whilst their priority is, of course, lower than that of the "base".

Just as is the case with the card used to make up a hand, players can use both their own hole cards or community cards to draw a Kicker from. The only thing that matters is its numerical rank. 

If several players have fully identical hands, including all usable Kickers, they are considered equally strong and get to split the pot.

Examples

Here, both players have played a same-ranking One Pair hand, so the winner is determined by the Kicker. As you can see, player 1 has a higher Kicker—the King. Hence, player 1 wins.

Here, both players have played a same-ranking One Pair of Tens, and both use the same community card—the Ace—as the high Kicker. However, player 1's low Kicker is a King, which is higher than what player 2 has; hence, player 1 wins.

Here, both players have played a same-ranking One Pair of Nines. Both of the highest Kickers available to both players are community cards. Hence, the third Kicker comes into play, awarding the win to player 1 because he has a Queen.

Here, both players have played a One Pair of Jacks, with all three higher Kickers on the table, drawn from the community cards (the Ace, the King, and the Ten). Since a hand always consists of 5 cards, no more and no less, players in this case get to split the pot (regardless of the fact that player 1's Seven is higher than player 2's Three!).

Here, both players have played a Three of a Kind of Fives, but the highest Kicker belongs to player 1 (the King), hence the win goes to him.

Here, both players have played a Three of a Kind of Sevens. The community Jack is the highest Kickers for both of them. But player 1 has a Nine among his hole cards, and this determines him to win in this case, acting as the second Kicker.

Now, remember how a Three of a Kind can only involve up to 2 Kickers, since each hand consists strictly of 5 cards, no more and no less? Here, both players have played a Three of a Kind of Queens, and both higher Kickers for both of them are, in fact, community cards: the Ace and the King. Hence, the pot is split between them.

Now, we have a One Pair of Twos on the table straight away. Each player also has their own Seven among the hole cards, and there is one more Seven on the table, too. This means that both players, in fact, have played a Two Pair of Twos and Sevens, but player 1 wins because of the higher King Kicker.

Here, players have both played a Two Pair of Queens and Kings, with 2 Kings and 1 Queen available from the community cards and each player having another Queen of their own. The common high Kicker—the Ace—is also on the table. Since only one Kicker can be considered for a Two Pair hand, the pot is split in this case.

Now, here we can see that none of the players can produce any hand at all, but player 1 has a Queen high card, which awards him the win this time.

Here's a similar situation; both players have no hands to play. So we look at the High Cards and see that it's the community Ace that they both can (and will!) use. The second-highest card—the King—is also a community one. We move further down the card ranks and get to the Queen of player 1's: so he wins.

No hands to make up here either. As we examine High Cards (or Kickers) one by one, we can see that all 5 of them will be from the community cards for both players. Therefore, they split the pot.

Again, no hands available! The community cards contain 4 of the highest cards in the game: the Ace, the King, the Ten, and the Nine. Hence, player 1's Eight from the hole cards ends up being the fifth (and determining in this case) Kicker that lands him the win.

Now, this is quite a rare river! A Four of a Kind of Sevens right there on the table. The winner is determined by a Kicker, and in this case it's the Queen that player 1 has; so he bags the pot, too.