Suppose you have the ability to take a chunk of text and construct a partial parse state from it. In the case of regexp matching, these partial parse states are functions mapping one state of the regexp matching automaton to another. You need one more thing: the ability to append two of these partial parse states, combining them into one. In the regexp example, this is just function composition. The key here is that this operation must be associative, and there must be an identity element: some partial parse state such that combining it with another state doesn't change anything. And its result must also be a partial parse state. This combination of parse states and an associative binary operation is called a monoid.

Once you have these conditions fulfilled, you can do all sorts of fun stuff. For instance, you can represent a string as a tree of chunks, and cache partial parse states at the nodes in the tree. That way, when you change the string, you can recompute the changed parse states in something like O(lg n) time, rather than going through and re-parsing the entire string. Or you can almost trivially parallelize your parser.

A week ago, I did exactly this: I had a language that needed parsing, and I wanted to incrementally reparse when I changed the (potentially very long) string, so I used a finger tree and wrote an incremental parser. It works beautifully.