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I know that this specific type of sentences has a name, but I can't remember it.
The sentences I'm referring to are the sentences that describe an action that is performed when saying the sentences, like for example:
Any clues ?
These sentences are referred to as performative utterances. As you mention in the question, performative utterances are those in which the utterance itself (the act of speaking) is part of the action that the utterance describes. So, the act of saying "I wish you a happy birthday" itself is part of the act of "wishing one a happy birthday", and thus it is a performative utterance.
The other main characteristic of a performative utterance, as I understand it, is that they have no truth value (though this is not a unique feature of performative utterances). In other words, "I wish you a happy birthday" and its ilk cannot be deemed true or false.
As jlawler points out in the comments, J. L. Austin's How to Do Things with Words is a good source for learning about performative utterances. Alternatively, consult any text on pragmatics.
