We like board games. Miss N. got her first set of games when she turned 2. She now has a rather large collection of good quality games.
–Bingo (aka Loto) with simple pictures or just colors. Bingo also helps with vocabulary. On the photo is our little N. playing Mini-Bingo (by Haba).
–"go away monster" (by Gamewright). This simple game is a variation of Bingo. And it can also teach little kids how to deal with the under-bed monsters - just shout "GO AWAY!"
–memory. For the youngest kids, start with 9 different cards. Arrange them on the table face up. Give your child enough time to memorize them. Then turn the cards face down. Ask her to find the cat, dog, etc. If the child is just learning a concept of same-different, you can help her by playing another version of the game. Divide cards into 2 sets. Give one set to the child and hold another one in your hands. Each player deals one card at the time, compare cards to see if they are the same.
–UNO is another great matching game. Just make sure to leave the "special" cards out.