Almost everything a CPU can do can be reduced to some sort of logic equation. In math and theoretical computer science the most common logic operations are AND, OR and NOT. From an electrical point of view NOR and NAND gates are far more popular.
In theory every logic operation can be performed using only NAND gates. In practice it is more useful to use other gates as well. This post will describe how the different gates can be implemented using transistors and why, for my purposes, NAND and NOR are preferable over AND and OR.