Thinking, Fast and Slow is full of illustrative experiments and examples that you can even try on yourself!
Kahneman introduces the main characters of the book, System 1 and System 2, which are actually personifications of a model used to describe how the human brain works.
System 1 operates automatically and quickly, with little or no effort and no sense of voluntary control. For example, when you see the face of an irate individual, you automatically determine the person is angry with this part of your brain function.
System 2 allocates attention to the effortful mental activities that that demand it, including complex computations. This system is invoked to find the product of 17 and 24, for example.
While both System 1 and System 2 are running while one is awake, System 1 is constantly analyzing the environment whereas System 2 is usually running at a minimal level. You can force your System 2 to activate fully by performing calculation-intensive tasks. When this occurs, your pupils dilate to such an extent that Kahneman is able to determine when an individual has given up on or completed a problem (as pupils contract to a measurable extent).
But System 2 has limited resources, as it cannot do too many things at once. For example, people who were asked to remember several digits were more (than others who were not subjected to such strain) prone to choosing an unhealthy food, demonstrating System 2's limited capacity. Kahneman interprets this to show that System 1 has more influence on us when System 2 is busy, and that System 2 can only juggle so many balls at once.
System 2 is also lazy. If we encounter problems that look easy, System 2 won't expend much strain, allowing System 1 to answer intuitively. For most easy questions, this produces adequate results; but for trick questions, for example, this lack of effort results in wrong answers.