A Markov chain is a set of states. You can move from some of the states to other states, and there's a certain probability of the direction you will take. For example, you might go for a walk regularly. After you're done walking, you either go home (75% chance) or go out for ice cream (25% chance). There are three states here - walking, home and ice cream. If I wanted to model your behavior, I could say "Okay, now they're walking. What do they do next?", roll some dice, and then decide if you went out for ice cream or went home after your walk. With Markov chain text generators, the states are just individual words, and you transition from one word to another based on the probability that those two words appear together.
Your explanation has made the most sense out of all the ones I've come across in the past month. If you wouldn't mind, to help my brain get a better grasp on this, in what classification/category does a Markov chain fall? Does it have parents, siblings, or strange friends with similar yet at the same time diverse interests?
Hm... I don't know more than the basics, really, but you can think of a Markov chain as being a directed graph (as in graph theory, not charts in Excel), with weighted edges like this. A and E are the two states, so if you're in state A, there's a 40% chance you'll move to state E, and a 60% chance you'll stay in state A. If I'm not mistaken, it's related to Bayesian inference as well, since they both address the same basic question (If I know that X is true, what's the chance that Y will happen). I hope that helps.