Yes, a Markov chain is a finite-state machine with probabilistic state transitions. To generate random text with a simple, first-order Markov chain:
- Collect bigram (adjacent word pair) statistics from a corpus (collection of text).
- Make a markov chain with one state per word. Reserve a special state for end-of-text.
- The probability of jumping from state/word x to y is the probability of the words y immediately following x, estimated from relative bigram frequencies in the training corpus.
- Start with a random word x (perhaps determined by how often that word occurs as the first word of a sentence in the corpus). Then pick a state/word y to jump to randomly, taking into account the probability of y following x (the state transition probability). Repeat until you hit end-of-text.
If you want to get something semi-intelligent out of this, then your best shot is to train it on lots of carefully collected texts. The "lots" part makes it produce proper sentences (or plausible IRC speak) with high probability; the "carefully collected" part means you control what it talks about. Introducing higher-order Markov chains also helps in both areas, but takes more storage to store the necessary statistics. You may also look into things like statistical smoothing.
However, having your IRC bot actually respond to what is said to it takes a lot more than Markov chains. It may be done by doing text categorization (aka topic spotting) on what is said, then picking a domain-specific Markov chain for text generation. Naïve Bayes is a popular model for topic spotting.
Kernighan and Pike in The Practice of Programming explore various implementation strategies for Markov chain algorithms. These, and natural language generation in general, is covered in great depth by Jurafsky and Martin, Speech and Language Processing.