1. modern linguistics as a change of paradigms

How many kinds of speech acts are there?

Some researchers have extended the classical lists of "speech acts" to include many actions that are felt to be helpful in analyzing task-oriented dialogs, things like "answer", "accept", "reject" and so forth. One influential set of ideas about this is expressed in the so-called DAMSL ("Dialog Act Mark-up in Several Layers") proposal.
For another, funnier take on an extended set of speech acts, listen to this scene (transcript here) from Chicago's Neo-Futurist group.

2. Conversational implicature

The work of H.P. Grice takes pragmatics farther than the study of speech acts. Grice's aim was to understand how "speaker's meaning" -- what someone uses an utterance to mean -- arises from "sentence meaning" -- the literal (form and) meaning of an utterance. Grice proposed that many aspects of "speaker's meaning" result from the assumption that the participants in a conversation are cooperating in an attempt to reach mutual goals -- or at least are pretending to do so!
He called this the Cooperative Principle. It has four sub-parts or maxims that cooperative conversationalists ought in principle to respect:
(1) The maxim of quality. Speakers' contributions ought to be true.
(2) The maxim of quantity. Speakers' contributions should be as informative as required; not saying either too little or too much.
(3) The maxim of relevance. Contributions should relate to the purposes of the exchange.
(4) The maxim of manner. Contributions should be perspicuous -- in particular, they should be orderly and brief, avoiding obscurity and ambiguity.
Grice was not acting as a prescriptivist when he enunciated these maxims, even though they sound like prescriptions for how to communicate. Rather, he was using observations of the difference between "what is said" and "what is meant" to show that people actually do follow these maxims in conversation. We can see how this works in considering the maxim of quantity at work in the following made-up exchange between parent and child:

Parent:	Did you finish your homework?
Child:	I finished my algebra.
Parent:	Well, get busy and finish your English, too!

The child did not say that her English homework is not done, nor did she imply it in a legalistic sense. Nevertheless the parent is likely to draw this conclusion. The implicit line of argument is something like this: the child would have simply said 'yes', without mentioning any particular subjects, if that answer were true; the fact that she referred to algebra, and did not mention other subjects, suggests ("implicates") that the unmentioned subjects are not done.
Very often, particular non-literal meanings are conveyed by appearing to "violate" or "flout" these maxims. If you were to hear someone described as having "one good leg", you would be justified in assuming the person's other leg was bad, even though nothing had been said about it at all.
A more elaborate taxonomy of types of conversational implicature, with illustrative examples, can be found here.
[Exercise for the reader: the last sentence implicates that you should go read the referenced site. What kind of implicature is that?]
Recent work in Relevance Theory builds on Grice's insight about the nature of communication:
"Relevance theory may be seen as an attempt to work out in detail one of Grice’s central claims: that an essential feature of most human communication, both verbal and non-verbal, is the expression and recognition of intentions. In developing this claim, Grice laid the foundations for an inferential model of communication, an alternative to the classical code model. According to the code model, a communicator encodes her intended message into a signal, which is decoded by the audience using an identical copy of the code. According to the inferential model, a communicator provides evidence of her intention to convey a certain meaning, which is inferred by the audience on the basis of the evidence provided. An utterance is, of course, a linguistically coded piece of evidence, so that verbal comprehension involves an element of decoding. However, the linguistic meaning recovered by decoding is just one of the inputs to a non-demonstrative inference process which yields an interpretation of the speaker's meaning."