Hidden Markov models (HMMs) are latent variable models based on the Markov chains. HMMs are widely used especially in speech recognition and there is an extensive literature on them. The tutorial by Rabiner  is often considered a definitive guide to HMMs in speech recognition. Ghahramani  gives another introduction with some more recent extensions to the basic model.
Before going into details of HMMs, some of the basic properties of Markov chains are first reviewd briefly.