In a previous comment, I sought to further understand how the Lottery possesses the Markov property. Based on your definition above, I can see that it does simply because the distribution X_t of winning numbers has the same dependence on X_{t-1} as it does on X_1, ..., X_{t-1}, that is, zero. Do I have that correct?
For an example without the Markov property, consider the sequence of random variables X_1, X_2, ... with X_1 being either -1 or 1 with equal probability, and X_t being normally distributed with mean X_1 and standard deviation 1.
Knowing the history X_1, X_2, ..., X_{t-1} gives you the exact distribution of X_t (since you know X_1), while only knowing X_{t-1} gives you much less information. This fails to be a Markov chain because the state X_{t-1} doesn't "remember" which of the two possible distributions is being used.
Edit: Wikipedia, of course, has a good overview of applications of Markov chains. http://en.wikipedia.org/wiki/Markov_chain#Applications
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.