What are the Cognitive Functions, exactly?

At its core, Cognitive Function theory describes how we process information (with Sensing and iNtuition) and make decisions (with Thinking and Feeling). These processes are then largely performed either externally or internally (Extraversion or Introversion), which creates eight different Cognitive Functions. A unique order of four of the functions creates your MBTI type.

If you want to learn more from experts (and not just from an artist on the internet), I find the type descriptions at Type in Mind to be very thorough and interesting.