Austin develops the notion of an 'illocutionary act' from his earlier doctrine of 'performative utterances'--expressions which, roughly, defined as not being true or false, but rather being uttered in the performance of an illocutionary act. According to Austin, these 'performative utterances' serve the performance of what he calls an 'Illocutionary Act'. 'Illocutionary acts', in turn, are later defined by means of two characteristics: they (a) are what is today called "institutional acts" ("conventional acts", Austin would call them); and (b) for the performance of the act the 'securing of uptake' is required. (The 'securing of uptake' amounts to the achievement of the understanding, in an audience, (i) that a certain act is (being) performed, and (ii) which act that is.)
That this is really Austin's notion of 'illocutionary acts' in (1962) can be derived from his list of the effects with which the 'illocutionary Act' is connected (1975, 116-7), from the summary of the discussion of this issue (1975, 121), and from his subsequent application of the relevant criteria of being an 'Illocutionary Act' to the act of stating (1975, 139).
A more comprehensive characterisation of those institutional acts, though an indirect one, is given in the first part of Austin's study, where he still concentrates on the investigation of those 'performative utterances'--which he defines as expressions whose utterance is part of an Illocutionary Act (cf. 1975, 6n2): "(A.1) There must exist an accepted conventional procedure", he writes, ...
... having a certain conventional effect, that procedure to incude the uttering of certain words by certain persons in certain circumstances, and further,
(A.2) the particular persons and circumstances in a given case must be appropriate for the invokation of the particular procedure invoked.
Furthermore, ...
The procedure must be executed correctly [...] and completely. (1975, 14-5)
The examples from which Austin used to start are marrying someone, naming a ship, bequeathing a watch, and making a bet--all these are most plausibly characterised as institutional acts for whose performance the 'securing of uptake' is required; characterisations of them as mere acts of 'meaning something (in any of the different 'speaker meaning' senses), or 'saying something', or 'communicating something', would quite obviously be inadequate: of all the definitions at hand, Austin's own, according to which the 'Illocutionary Act' is an institutional act, is clearly the only one that fits to this choice of examples.
Since the term 'Illocutionary Act' was subsequently used to refer to numerous phenomena somehow connected with linguistic meaning, it is worth mentioning that Austin himself speculated (perhaps following Wittgensteinian ideas) that these acts might somehow be essentially connected with language. In particular, he speculated that 'performative utterances' might be subject to a grammatical peculiarity. Yet his painstaking search for such a criterion is unsuccessful, as he clearly confirms himself (1975, 60, 63-4, 91).
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Austin, J.L. (1975), How to Do Things With Words (2nd edition), Oxford: Clarendon.