In JL Austin’s theory of speech acts, an illocutionary act is any utterance by which the speaker performs a certain action. Examples of such action can be an argument, a question, a promise, an order, an apology etc. in other words, an illocutionary act is the act performed by the act of speech.
The illocutionary act includes the locutionary act, for example the sentence “it is raining”. But it becomes an illocutionary speech act when it is performed for example in order to alert someone to the fact that it is raining outside, thus affecting his considerations or actions.
In other examples the illocutionary act can contradict the locutionary act. Take for example the statement: “I am dead tired”. The speaker does not mean the he is actually dead, but wants the listener to understand the he is extremely tired.
In order to give good examples of illocutionary acts take the meaning ingredient of a locutionary act and add the aspect of a receiver of that meaning, that is the object of the speech act.
see also: illocutionary force