Beginners ask me this more than any other rules question, usually right after a game where both kings ended up chasing each other around an empty board. So, the short version: a king cannot capture a king, and it cannot even try. The reason is not a special rule about kings fighting. It falls out of the ordinary rule for how the king moves.
The rule that makes it impossible
A king may not move to a square that is attacked by an enemy piece. That is the whole law of king movement, and it does a lot of work. A king attacks the eight squares around it. So the square next to a king is, by definition, attacked, and your king can never step onto it.
Look at the board. White’s king on e4 would love to walk to d5, e5 or f5. Every one of those squares is covered by Black’s king on e6, so every one of them is off limits.
Put it the other way round and you get the answer to the question. To capture something you have to move onto its square. A king can never reach a square next to the other king, let alone the square it stands on. The two kings spend the entire game with at least one square between them, a standing arrangement that players call the opposition when it matters in an endgame, and there is a page on what the opposition is because it decides a lot of king and pawn endings.
The game ends before any king is taken
There is a second reason the capture never happens, and it is the reason you will never see any piece take a king, not just a king. Chess ends at checkmate. Checkmate means the king is attacked and there is no legal way to stop the attack. The capture itself is never played; the rules stop the game one move early, which is why you never actually capture the king with anything.
Here is what that looks like with the simplest mate there is. White’s queen on g7 is next to the black king on h8, and White’s king on f6 guards the queen. Black’s king is attacked and has nowhere to go. Game over, and the black king is still standing on the board.
8 
Notice the white king’s job in that picture. It is not there to capture the black king. It is there to protect the queen so the black king cannot capture her. The king is a fine attacking piece at close range, as long as it keeps its one-square distance from its opposite number.
Where the confusion comes from
Three places, in my experience teaching this.
The first is online chess, where the software simply refuses the illegal move and the player never finds out why. Set up two kings on a real board and try to walk one into the other and the rule becomes obvious in about ten seconds.
The second is the phrase “take the king” that people use to mean win. It is a figure of speech. The King of the Hill variant, where you win by walking your king to the centre, does not change it either; you still cannot capture the other king there.
The third is a genuine edge case: a king can deliver check, just not with itself. Move a king off a line and a rook behind it can suddenly be attacking the enemy king. That is a discovered check, the checking piece is the rook, and the kings are still not touching. Likewise a king can capture an undefended queen or rook standing next to it, and does so all the time in endgames; there is a page on whether a king can capture a queen. The only piece the king can never capture is the other king.
What to do with this in your games
Two practical things. First, when you are winning with a queen or a rook, use your king. It cannot be captured by the enemy king, so it can walk right up to one square away and take away escape squares. Every basic checkmate is built on that. Second, when you are the one being hunted, remember that the enemy king cannot come closer than one square. If the only attacker left is a king, you are not in any danger at all, and in fact a lone king can never win.
And if you want to see the rule in action rather than read about it, mate-in-one puzzles are a good place to start: in every one of them a king is attacked, cannot escape, and is never captured.