Cognition--from which "cognitive task" derives, is the scientific term that essentially means "the process of thought." Cognitive tasks therefore, are the actions which help define cognitive thought process.
Usually the term "cognitive task" is used in the context of "cognitive task analysis." This is the process of analyzing or testing what level of thought process a person has reached or is using to solve a specific problem. You might look at a toddler, for example, who touches a hot iron and gets burned. If this toddler learns not to touch the hot iron again, and actually avoids it, he has demonstrated the cognitive task of understanding the iron will hurt him and not to touch it.
In elementary education, for example, it is common to measure cognitive tasks in subjects like mathematics. Addition and subtraction are learned and understood through a series of cognitive tasks which build on one another. A 1st grader might be able to handle physically counting 3 objects, adding one to the pile, and counting 4 as the result. Gradually, the physical object will also be represented by number symbols, and the understanding will shift from the physical addition of one object to the pile, to the mental understanding that 3+1=4. Each step in the process of learning 3+1=4 is a series of cognitive tasks.
Many cognitive tasks are linked directly to short and long term memory. Others are developed physically (as though a process of motion). Humans learn and build understanding though hearing, speaking, and doing. One theory utilized in elementary education (and even higher) is the idea that blending these three learning styles/techniques as much as possible allows students to experience and attempt multiple cognitive tasks, providing the most opportunity to establish understanding.