A small technical precision about the definition of HMMs : in addition to the matrices defining the transition probabilities (usually called "A") and the symbol emission probabilities (similarly, matrix "B"), a third matrix is often provided that gives the probabilities for each state to be the initial one (instead of just giving one initial state). That third matrix (a vector, actually) is usually referred to as pi.
Hidden Markov Models are also used quite a lot in computational linguistics, for example to solve the ambiguities when tagging a piece of text (that is, saying "this is a noun, this is a verb", etc.) : when encountering the word "green", you can't tell whether it's a noun (the colour) or an adjective -- so you associate it with the symbol Noun|Adjective, which can be emitted either by the state Noun or the state Adjective. For that particular application, people sometimes refer to the symbols as "ambiguity classes". Apart from that, the principle is the same as usual with HMMs : finding the path in the network that gives the best score for a given sequence of symbols, and keeping the states (i.e. the tags) found on that path. With a proper training and maybe a few additional heuristics to help the tagger, it's quite easy to reach an accuracy of about 95% -- which is an excellent result when you deal with natural languanges.