The pre-operational and concrete operational stages of development in humans are tenets that belong to Jean Piaget's theory of cognitive development.
Piaget's theory argues that there are specific moments in the development of individuals in which particular processes enable (and sometimes disable) the acquisition of new information and knowledge.
These processes occur in tandem with age. In other words, as a specific age is reached, a process is supposed to take place as the individual develops. The pre-operational stage, for example, is the first stage of Piaget's theory of cognitive development. It occurs from ages 2-7 and it is expected that children of this age group would start creating representations of objects mentally, tend to pretend-play (which is also known as the semiotic functions) and will develop a form of egocentric behavior which is typically more evident at age 2 (the terrible two's). This behavior is characteristic for the child's unwillingness to share, or partake from attention aside for attention given to the child himself.
Conversely, the concrete operational stage of development pertains to children within the age group of 7-11 years of age. It directly follows the pre-operational stage and, at this point, children are beginning to create ideas, build schema, and categorize emotions, thoughts, and beliefs. With these cognitive tools, children are able to think somewhat logically, or at least inductively. They are naturally curious, make lots of questions to guide their learning process, and they can perform concrete activities. It is, however, difficult for children of this age group to conceptualize the abstract, to use mental processes without manipulatives, or to use deduction to come up with answers.
Those are the ideals and outcomes that are inherent to the pre-operational and concrete-operational stages of Jean Piaget's theory of cognitive development.