I take that back... I seem to have been informed wrong. These days they tend to use bigger arrays and other systems. More info here:
http://www.top-5000.nl/ps/SomeAspectsOfChessProgramming.pdf
And here:
http://en.wikipedia.org/wiki/Board_representation_%28chess%29
Dude! Thank you. :)
I don't agree that you need to begin with something like Tic-Tac-Toe first. Don't learn how to program by reading a book and not implementing examples; rather, choose an ambitious project and look up what you need to know. For example, let's say you choose to do it in Java (which I would strongly recommend: It would be a great introduction to OOP). You realize that you need some way of representing a board. Boards are two-dimensional, so let's see if there's a data structure that allows for this. Hey, 2-d arrays! It's a much better way of learning than reading it in a textbook, because it's sometimes difficult to see the purpose.