Norris' Markov Chains is a good starting point (you can read the first chapter from the author's website). I think this is the best introductory text that I know of.
Assmussen's Applied Probability and Queues is another good book.
Meyn and Tweedie's Markov Chains and Stochastic Stability (1993 version, current version is significantly updated) is a nice book, but not so explicit. See also Meyn's Control for complex networks text (has a link to his "crash course on markov processes" as well). [Free from author's website]
There is also a bit of material in normal probability books (e.g. Durrett's Probability: Theory & Examples)
Markov chains have an extremely wide range of applications, so there are a lot of books you can look at -- you may want to be a bit more specific on what you want to know about markov chains.