One of the oldest games in the book is Memory.
It is a gem. It is flexible. It is an activity that is easy to personalize so that it fits your child’s interests. It teaches many concepts at different levels. You can play this game with really young children, but you need to simplify it so it is not overwhelming.
All you need: 2 sets of stickers, 3 x 5 cards, and scissors.
All you do: Cut the cards in half, then place the stickers on the cards. 
For children who are between 2 1/2 and 3 years just begin by matching pictures. Put out an array of 6 pictures. Hold up a matching picture and say “Can you find the….put your finger on the ……” It can look something like this:
Once your kiddo understands the concept of matching. Set up a simple array that stacks the deck for success. Place the matching pairs beside each other. Turn them over, face down. You go first and show them what to do. Here is how to stack the deck:
Once they get the hang of this, add more pictures, then mix them up so that the difficulty is increased. Like this:
I suggest that when you increase the number of cards and use a random layout, you first match pictures face up. Once they understand the idea of finding the same pictures then turn the pictures face down. Placing the cards face down adds another level of complexity. This adds the memory component, it also requires more attention and focus.
Look at what you can teach your child:
1. Same/different
2. Vocabulary ~ animal names for this example.
3. Animal sounds ~ You can’t play a game with farm animals without saying ‘moo’, ‘oink’, ‘quack’, or ‘peep-peep’!
4. Attention to task
5. Turn-taking
6. Social skills that are associated with playing games.
7. Increased memory of course!
You can cover the cards with contact paper or have them laminated. They don’t have to be perfect. Mine aren’t, I don’t have time to make sure every cut line is perfectly straight. I would rather you spend time with your child then fretting over perfection.
Have fun!