Games for infants involve stimulating the child to interact in a range of ways. These are not really 'games' so much as repetitive activities that develop a child's understanding of the world. This can involve any play that encourages interaction between you and the child according to a simple rule of cause and effect. Simple games like peek-a-boo help babies to understand the concept of object permanence, the idea that objects and people do not disappear when they are out of sight. Try action-reaction games. Offer the child your two hands and make a different, funny noise when each of the hands is touched.
Children gain increased understanding of their environment when they interact with objects around them. Give them collections of objects that can be banged together or rubbed to produce different noises. This helps young children to understand the properties of things in the outside world and how they relate. Demonstrate and encourage imitation. Try variations of this game such as looking at what different items do in water: sink or float, hold or leak.
Older toddlers can be encouraged to think about categorization by presenting them with a collection of objects and helping them to sort them into groups. You can start with very simple groupings such as color. With older toddlers, you can move on to more complicated categories such as the room the objects belong in. Further develop this game by playing odd-one-out games, asking the child to pick which object does not fit with the group.
Older toddlers can hone language recognition and thinking by racing against you to accomplish simple tasks using simple objects. Say "Put the red shoe on the green chair" and encourage your toddler them to do it as fast as possible. This will help her to understand different ways of connecting objects and to quickly process communication.