I am letting you know now, Piaget's theory of Cognitive Development were quite flawed and he was not looked upon very highly for the idea that people do not develop cognitively past twelve.
But here it goes...
1. Sensorimotor Stage (Birth- 2 years): This is the stage in which the child experiences their world through their senses. Babies begin to learn object permanence (meaning they know that even if they can not see an object- it is still there.) The fact that they are living and learning through their senses explains why babies are constantly preoccupied with stuffing non food items in their mouths. The child also displays simple reflexes.
2. Preoperational (2- about 7): The child begins to experience their world through language. For example they may start to ask questions such as, why is the sky blue? How do birds fly? Why don't cats like dogs? This stage has the problem of egocentrism, the failure to understand conservation and the lack of understanding of people permanence. This is when Seperation Anxiety sometimes sets in because they believe that if mommy is gone she is simply gone- not coming back. They also experience animation- meaning they believe objects have life or emotions.
3. Concrete Operational (7-11): Children in this stage acquire a variety of new skills they previously could not perform. They learn seriation (meaning they can categorize objects by two or more traits...) If you give a child in this stage a bunch of star and heart shaped beads of every color, you could ask them to seperate them by shape and color at the same time and they would be able to do it. They learn basic math skills, learn to understand conservation, reversibility and they eliminate their previously instilled egocentrism. They also learn to draw basic conclusions from a set of information or from interpreting a situation.
4. Formal Operational Stage (12-Death): Abstract thinking and higher level math.