A poker hand must be comprised of five cards, so kickers can only come into play during certain types of hands like: Four-of-a-Kind, Three-of-a-Kind, Flush, Two Pair, One Pair, or High Card. It's important to note that kickers do not factor in to the following hands: Royal Flush, Full House, Straight, or Straight Flush.
The kicker itself is the leftover card not used to form the hand. So, for example, in a Four-of-a-Kind or Two Pair situation, four cards are used to form the hand itself, and one card is 'left over' (i.e. not part of the Four-of-a-Kind or Two Pair).
Let's say you have two players sitting at a table. Player One has the 7 of Clubs and 10 of Hearts as pocket cards, and Player Two has the 10 of Clubs and 9 of Spades as his pocket cards. On the table are the 10 of Diamonds, 2 of Spades, 4 of Hearts, 5 of Clubs, and 2 of Diamonds.
Given that situation, both of the players have Two Pair: Tens and Twos. Since only one player can win, we have to look at their 'left over' or 'kicker' card. Here's how their hands look:
Player 1: 10♥, 10♦, 2♠, 2♦, 7♣
Player 2: 10♣, 10♦, 2♠, 2♦, 9♠
Player Two's kicker (the 9 of Spades) is higher than Player One's kicker (the 7 of Clubs), so Player Two is awarded the hand and the entire pot.
Now, had the table cards contained the King of Clubs rather than the 5 of Clubs, the users' hands would have looked a bit different. Instead, they would have been:
Player 1: 10♥, 10♦, 2♠, 2♦, K♣
Player 2: 10♣, 10♦, 2♠, 2♦, K♣
In this case, the kicker is the same for both Player One and Player Two, so their hands are identical and they split the pot.