Why would you want to do this? So that your child has a fun thing to do that exercises the mind at the same time.
Young children learn better than you or me. Look how fast they pick up language from a standing start. You tend to get prodigies in maths, music and chess - I think it is because of pattern recognition. I won't develop that theme; you know what a good learner your child is.
- Okay, first you need to know and appreciate the game yourself. This being the case, I recommend that you let the child play with you over the board without the moves having to mean anything - but your shots should be legal (ignore check). Let your child take a shot with their own colour pieces, and alternate their shots with yours. They'll soon see that pawns, knights and the king tend to move a short distance. Your child will copy you. Name the pieces as they are played.
- Next stage (my daughter was four - a bit young maybe). Encourage the child to play legal shots (ignore checks). I found explaining the knight's move took forever so take my advice and don't push this one: let it sink in over time. Encourage them to win, 'Look - you could take my king from here! And I can't get it out of the way'. You are giving your child little successes and she will pick up the point of the game.
- The child will want to win all the time. Let them! Allow lots of quick checkmates. Keep games short by encouraging rapid development of pieces in the opening. Leave that Closed Ruy Lopez in the book for a bit. No place for an equal struggle here. Your child's pieces need to be develped centrally and yours, well, yours don't.
- Two ideas for developing an understanding of the game: play with just kings on the board and give your child both rooks. Play out the (end)game giving prompts when you need to. You should always lose and lose quickly. Try it out with king, queen and rooks versus king. You get the idea. And then...
- Play The Mad Queen. This is where one of you has the queen on its starting square, and the opponent has the standard array of pawns. If Pawn player gets one of their number to the back rank- Pawns win. If Queen takes all of the pawns before they can reach the back rank - Queen wins. This is a very good game for slightly older children. It only requires a basic understanding of how the queen and pawns move and teaches a huge amount about piece placement, double attacks, maintaining the initiative etc. Swap sides of course so that you get to play the pawn storm. Allow yourself to win a few games now and then because ...
Your child will be beating you at chess any day soon.