Your hand is made up of the best five card combination out of any of the two cards in your hand and the five on the board so you can use both, one or none of your cards in your hand*.
Your "kicker" usually
comes from your hand because the community cards are shared but it doesn't have to be in your hand. Note that "kicker" is just the term we use to describe the extra card that's hanging around after you've taken pairs and other cards into consideration. If you're having trouble understanding the concept it's best not to think in terms of kickers at all - just work out what the best five card hand each player can make is and it should become pretty obvious.
Player 1: AK
Player 2: A7
Player 1's best hand is AA99K. Player 2's is AA99Q. Both have aces and nines but player 1 has a king kicker and player 2 has the queen on the board. The king wins because AA99K is the best five card hand.
Player 1: AJ
Player 2: A7
Player 1's best hand is AA99Q. Player 2's hand is AA99Q. Both have the same hand with the same kicker so the pot gets chopped. The fact that player 1 also holds a jack, which is higher than player 2's seven, is irrelevant because you only play a five card hand and in this case the "kicker" both players use is the queen on the board.
Feel free to post any example hands that are confusing you and I'm sure we'll be able to clarify them. But in any case, just write down what the best five-card combination each player can make is and it should start to become obvious.
* Note that this applies to Texas hold 'em only. Omaha and other games have different rules about how many cards you have to use from your hand.