For instance, if I say, "Let me pay your bill," in a successful attempt to persuade you that I am kind, the locutionary act is the actual utterance as understandable to an English-speaker. The illocutionary force is the offering to pay -- I'm not merely saying the words, I am saying the words as an offer of payment -- and the perlocutionary force is persuasion -- I'm not merely saying the words, nor merely offering to pay, I am also succeeding in the attempt to use the offer of payment to persuade you of my kindness. You can think of them as the layers of our use of language: we communicate with words (locutionary), those words are said as a kind of act (illocutionary), and in the attempt to achieve further results by means of that act (perlocutionary).
When people discuss this, they generally stick to talking about particular statements, but there are other kinds of speech acts, including speech acts that have statements as component parts. A formal speech, for instance, is itself a speech act, and has its locutionary, illocutionary, and perlocutionary aspects, just as much as its component statements do. And the same is true for arguments. An argument has a locutionary aspect (the words and syntax that mean something); the argument is communicated in an act of a certain kind (e.g., as something argued for the sake of argument, or as a suggestion, or the like); and the argument is communicated in that kind of act in the attempt to achieve a result (e.g., to persuade you to agree with the conclusion, or to enlighten you as to someone's view, or to impress you with the sophistication of the argument, etc.). Historically, the illocutionary aspect has sometimes been called the dialectical part and the perlocutionary the rhetorical part.
It's very important not to confuse the illocutionary and the perlocutionary aspects of the argument, the dialectical and the rhetorical, although both can be important. The perlocutionary force -- like persuasion -- is not part of the immediate act itself but of a larger act subsuming it. You can succeed illocutionarily in the act itself, but perlocutionary success requires other things that are beyond the argument. In statements, I can promise by saying, 'I promise to do it'; I can't persuade you by saying, 'I persuade you to do it'. (That sounds like a punchline for a joke.) Arguments are no different. You can refute someone just by building an argument of the right structure on the right premises; you can't persuade someone just by building an argument, because even if you are trying to persuade someone, persuasion is a further effect in another person that you are trying to achieve by the refutation. The success conditions are entirely different.
One reason for keeping the three-act theory is that it is essential to Austin's notion that communicating with language is a practical activity, and therefore has the structure of a practical activity. Throughout discussions of practical activities one finds analogues of a three-act theory. For instance, in moral theory, if you hit someone in the face, there is the act itself (the 'locutionary' aspect), the object in the act (the 'illocutionary' aspect), and the intent of the act, that is our use of it for further ends (the 'perlocutionary' aspect, or at least the perlocutionary aspect it will have if it is successful). For instance, you might be hitting them in the face as retaliation for the fact that they look better than you in the attempt to harm their face; or you might be hitting them in the face as a form of self-defense in the attempt to deter them from doing something bad. It matters greatly that you can in some sense do the same action with a different object; entire ethical theories, like Kantianism, are built on the point.