The analysis/reasoning core is what people often think about when they talk about AI. It is the part that actually uses the sensory data and the memory to analyze the current configuration and make a decision. Popular methods for such tasks are finite state machines and rule systems. Making a decision can be slow or fast depending on the number of alternatives and the sensory data to evaluate. Chess playing is a slow process, whereas moving a character in Quake is really straightforward. Obviously, a character in Quake has a limited range of options (usually, moving in four directions, jumping and shooting), whereas 20 to 50 moves can be made on a chessboard, given an initial configuration.
Luckily, many games require only simple decision-making processes involving a few choices, and great results often come at a relatively low price. As you will soon see, a lot of games praised for their great AI have been built with relatively simple algorithms. Thus, let's now move on to specific technologies.