The term "prose" is now generally understood in the same way as described by the Maître de philosophie in Le Bourgeois gentilhomme, i.e. language that "sounds natural and uses grammatical structure" and "the opposite of verse" (quoted from LiteraryDevices.com). Hence, on the surface level, any normal language that is not verse is prose. According to the Maître de philosophie in the play, this includes oral language (since he appears to agree with Mr Jourdain's discovery that he is speaking in prose). However, some definitions of prose exclude spoken language, e.g. LiteraryTerms.net, which says, "Prose is just non-verse writing".
Depending on which definition of prose you accept, the joke in Le Bourgeois gentilhomme has three levels:
- Mr Jourdain discovers that he uses normal language that is not verse. This is hardly something to be proud of, but he is impressed that this deserves to be called "prose". (See for example, the emphasis on "prose", when Jourdain says, 'Quoi ? quand je dis : « Nicole, apportez-moi mes pantoufles, et me donnez mon bonnet de nuit », c’est de la prose ?': https://youtu.be/jJMuzRxE5sA?t=419.) This is the joke that people allude to when they say they are like Mr Jourdain, i.e. they have been doing something that is perfectly ordinary and then discover that there is an impressive term for it.
- Mr Jourdain discovers that his ordinary language is called "prose", which is a term that is usually reserved for literary language that is not verse. (As mentioned above, there is no consensus about this.) According to this interpretation, both Mr Jourdain and the "Maître de philosophie" look ridiculous, because they don't understand the difference between ordinary spoken language and literary prose. (The "Maître de philosophie" is also a comical figure, after all.)
- Finally, "prose" can also mean language that is banal or commonplace (the English Wikipedia article does not mention this, but the French one does). Thus, Mr Jourdain's pride at discovering that he has been speaking prose for forty years without realising it makes him look doubly ridiculous: from his own point of view he is proud of doing something that is worthy of the literary terms "prose", while all his life he has been saying utterly banal things. In fact, he even provides examples (see the quote in the first list item). Most readers and viewers of the play never get this double (or triple) entendre.