In teaching chess to children, one of the most difficult concepts to explain is that of the illegal move. In chess, there are some moves which simply aren't allowed, not because they are bad moves, but simply because they are against the rules. Of course, it's simple to understand the most basic of illegal moves in chess; a rook can't move like a knight, for instance, and if you try to do so, it's illegal.
The difficulty usually comes in understanding illegal moves involving the king, and how to react to them properly. The most important thing to remember is that a king must never be moved into check, or left in check. If a player makes a move that exposes his king to check, that move must be taken back, and a legal move must be made instead. If a player is in check and makes a move that doesn't remove his king from check, again, that move must be taken back and a legal alternative found instead.
Now, here's the part that often trips up new players: if a player doesn't notice this, and leaves their king exposed to a check, the opponent is not allowed to capture the king! It is as much their responsibility as the player's to point out the illegal move and allow the player to make a legal move instead.
Another related point is that two kings can never be placed on adjacent squares. The reasoning for this should be obvious: if one king is next to another, whoever is on the move would be able to capture the other king! Kings are very good at assisting in checkmates (by preventing the other king from moving near them), but they can never get close enough to actually attack the other king.
Understanding illegal moves and how they effect the king is one of the most challenging hurdles facing beginners learning chess. It's not intuitive that even though the goal of the game is to capture the enemy king, you must allow your opponent every opportunity to protect their king by legal means. Once a new player completely understands the difference between a real checkmate and "taking the king," they likely understand all the important rules in chess and are ready for more advanced study.