Here is a simple game that you can play with your child(ren) that basically teaches about rolling dice, number recognition and taking turns.
I came up with this game after a failed attempt to play a simple homemade board came with my then two year old. It had originally consisted of spinning a die (a d6) and then moving your piece (minis, in this case) around a hand drawn track. This had problems with a two year old because the concept of 'winning' is still not important to her. So I simplified the game to: Tickle Six.
The game is played thusly. You will need:
A d6 (otherwise known as a 'die')
Sit in some kind of formation. Take turns rolling the die. If you roll a six, you get tickled.
With this game, my daughter learned about taking turns, recognized numbers and got to be tickled. The turns are shorter which is good for a toddler but still exist so they learn to take them. The easiest number she recognized first was 'six,' of course but through continued play and calling out the numbers while playing, she learned the rest of them. I'm not a child rearing/psychology expert but I know she has fun playing it. She still wants to play it and I've play tested it on a 12 year old, who also enjoyed it.
Please be sure your kid is not one of those 'ticklers' that actually just rips chunks of flesh from people or it will not be fun.