It's nice to see an explanation of Markov chains (and some techniques for predicting their behaviour) targeted at a wide audience. My only complaint is that the provided definition of a Markov chain is somewhat unclear and imprecise (as pointed out already by lambdaphage). An alternative definition that I like a little better is given below.

A Markov chain is a sequence of random variables X_1, X_2, X_3, ... with the property that the distribution of X_t given the complete history X_1, ..., X_{t-1} is identical to the distribution of X_t given only the previous state X_{t-1}. Intuitively, this means that the state X_{t-1} sufficiently summarizes the history of the chain so that knowing the earlier states of the chain does not allow you to better guess the next state. This property is called the Markov property.

For those looking for a free and in-depth reference, I encourage you to check out Byron Schmuland's course notes: http://www.stat.ualberta.ca/~schmu/stat580/2012notes. He also links to several other free references for Markov chains on his course webpage: http://www.stat.ualberta.ca/~schmu/580.