The association areas of the brain are the last to develop (recall that one of the main causes of brain growth after birth is making new connections between already existing neurons), so human's cognitive ability lags behind their physical ability. Cognitive growth is driven by the desire to understand the world around us.
The main theorist regarding cognitive development is Jean Piaget. A Swiss, Piaget got his PhD at 21, which is about the age that most people now get their BA. He worked in Paris at Alfred Binet's laboratory school, where the first tests of intellectual ability were developed. Piaget was intrigued at how children answered questions incorrectly, as the mistakes that they made seemed to be consistently different from (the correct) adult responses. He made careful studies of his own children, and he expanded those studies to larger groups of kids. Piaget was only discovered by North Americans in the 1960s, when some of his work was first translated into English. However, he had been a big name in European psychology since the 1930s.
Piaget suggested that people create schemas, which are mental models of how the world works. Piagets postulated two general processes that governed schema change:
Assimilation, which is putting new things into old schemas
Accommodation, which is modifying existing schemas to fit new experiences
Piaget theorized that there are 4 stages of human cognitive development. He suggested that the stages occur in an invariant developmental sequence. However, he also suggested that the exact ages that each child will pass from one stage to the next can vary.
Sensorimotor (birth to 2 years)
According to Piaget, the first responses that infants make in the world occur by chance, so infants discover the reponses that they can perform by chance.
It is only between 8 and 12 months that infants begin to make intentional actions; here problem solving behavior first appears, and the child will perform actions as a means to an end.
Shortly after this period, children will begin experimenting with new ways of doing things that will satisfy their curiousity.
The result of that experimentation will be that children develop an ability for inner experimentation, where they can perform an activity in their head before doing it in the world.
Imitation ability beings during this stage as well, and by the end of the sensorimotor stage (18-24 months), children can perform deferred imitation, where the child imitates an actor seen sometime earlier.
Others have suggested that the ability to perform deferred imitation appears earlier. There is evidence that 14 month olds can imitate the actions of live models as much as a week after seeing them.
Object permanence is also a part of the sensorimotor stage. Object permanence is the idea that an object is permanently in the world, even if it is removed from view. Infants start out believing that objects placed out of sight are out of existence, and gradually acquire the belief that out of sight does not mean out of this world.
Recently, the time when children develop object permanence beliefs has been questioned - by studies such as the one involving the car and ramp - as well as other beliefs regarding what Piaget suggested developed during his sensorimotor stage of development.
One suggestion as to why Piaget got the timing of abilities wrong is that he relied too much on motor responses as evidence of knowledge, and much of the evidence inconsistent with his beliefs is purely perceptual learning without much (if any) motor responses.
Preoperational (2 years to 6 years)
Infants start life as egocentrics, as they are unable to think of another person's point of view. This egocentrism can explain why children may appear selfish, as they are unable to think of others viewpoint. The notion of egocentrism is related to the idea of whether children can think symbolically. That is, when can a child imagine the other person as another agent in the world, or symbolically think of the other person's actions?
Conservation is another notion that children must learn. Initially, they judge objects on their size rather than content, so changes in the shape of a container will affect their judgment of which is bigger rather than the actual volume. Examples: beads in a row, balls of play-doh, tiles filling up a square.
Related to the conservation idea, children also do not always distinguish between appearance and reality during this period. The appearance/reality distinction refers to the ability to keep the true properties of an object in mind even when the appearance of the object changes. The cat/dog example with 3 year old and 6 year olds.
Concrete Operations (7 years to 12 years)
Piaget proposed that children being thinking mentally during this stage. That is, symbolic thinking begins during this stage. For example, given concrete materials, children begin to understand that materials don't change quantity when they change shape. Piaget thought that children in this stage could do many abstract mental operations on objects that were physically present, but that they couldn't reason about objects that were not physically present, such as the Xs, Ys, and Zs used in algebra.
However, children seem to develop symbolic thinking earlier than Piaget proposed. Judy DeLoache at the University of Illinois did the Snoopy studies, where 2 year olds were asked to find a big Snoopy doll in a big room after being shown tiny Snoopy in a tiny room; they couldn't find the toy. However, 3 year olds will immediately find the big Snoopy doll.
Formal Operations (12 years onward)
Here children begin to be able to reason formally and abstractly, and can do so with purely abstract and symbolic things.
One problem that highlights the differences between children who are have not reached this stage and children who have is attempting to figure out why a pendulum swings faster or slower. There cause could be either the weight of the pendulum, its initial starting force, or the length of the string it is on. Children who are not yet in the formal operations stage will vary more than one of these factors, which make accurate conclusions difficult, but children in the formal operations stage will vary only one factor at a time - holding the others constant - to examine exactly the effect of each factor on the pendulum's motion.
Reasoning such as that necessary to answer the pendulum problem is known as hypothetical-deductive reasoning, and it is a general method that can be applied to systematically solve any problem. Some people have suggested that this type of reasoning is used in science.