Briefly, Combinatory Categorial Grammar is an alternative to phrase structure grammar. In the simplest form, there are two syntactic categories, N(oun) and S(entence), and these will correspond to the semantic types Entity and Proposition. Complex categories are built by concatenating categories with a directed slash. S\N is a complex category that combines with a noun to the left, since the slash points to the left, and returns a sentence. In more common phrasing, this is (one possible) syntactic category for intransitive verbs. Each syntactic category is associated with a semantic category, in the lambda calculus, whose type is defined in terms of entities and propositions. The type of the semantic category for this proposition would from entitities to propositions, or <e,t> in Montague's notation. Modifiers typically have a syntactic category of the form X/X or X\X: they combine with the thing they modify but do not change the syntax.
There are other rules for combinations as well; for example, S/(S\N), an S looking for (S\N) on its right, can compose with (S\N)/N, (S\N) looking for N on its right, to yield the category S/N. This is useful for analyzing things like gapping.
Here is a good short introduction, and
this one also looks good. For a much (!) more comprehensive overview,
have a look at this paper.
Here are some things to keep in mind when reading about CCG. First, sometimes people include apparent simple categories in their derivations besides S and N. Sometimes they are just abbreviating long complex categories (writing TV for transitive verb instead of the much longer category (S\N)/N), while other times they have really added new simple categories. Second, different applications will consider different combinatory rules. In particular, theoretical papers often include lots of type raising and compositional rules because these expand the expressive power beyond a context free grammar, but papers with computational implementations often omit those rules so the grammar can be implemented and evaluated as a context free grammar.
