My conversation manual included the word assai very often as an equivalent for troppo, molto, etc. When I got in Italy one of my friends from there, of Neapolitan origin, said that is a Neapolitan word, not used in Italian and it is considered vulgar to use it, because of its peasant connotations. I heard it numerous times on TV and the shows weren't about history/ethnology, but nowadays subjects. Also other friends from Veneto didn't knew its meaning. So what's the story with this word?
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Your friend is wrong when he says that this word is a regionalism. It is usually used as an adverb meaning "much, very", intended to reinforce an adjective:
It may also be used in the sense of "many" in front of a noun (so, taking the value of an adjective). Beware, it's an old usage, mostly considered wrong nowadays (except than probably in the South):
Moreover, a third usage exists. You can indeed use it to mean "much" as an adverb:
All this is perfectly fine in Italian (and that's why you hear it very often).
This said, I understand (a bit) what your friend meant. In Naples and the neighborhood, the word is used after an adjective (in the same sense outlined in my first example here), and this is a regional usage. It is anyway wrong to say that it does not exist in Italia. If the one from Veneto didn't know the word, it's his/her fault.
As for the etymology, the word comes from ad satis, Latin, meaning "enough" and indeed its original meaning has been shifted from "enough, in a sufficient way" to "much" (you can read the Italian etymology dictionary here).
What on Earth makes them think it is “vulgar”? It is a regular, non-regional Italian word (it appears in Petrarca, Leopardi etc. and is used for instance even in musical notation: allegro assai etc.). As for many words, it is used more or less in different parts of Italy (or, for other words, in different strata of population, or by different age groups): this are matters studied by sociolinguistics.
A point to be careful about: Its original meaning is enough (from Latin ad satis), but it has been used for centuries in the sense of “much, very.”
(As always happens for the meaning of single words, most of the story can be read in the relevant article of any good dictionary.)
Assai is a sort of superlative and it is absolutely not vulgar. It comes from the Latin word satis, which means very or much depending on the usage. In Southern Italy, it is often used as an adverb with the same meaning. For example:
[Correct usage] Un uomo assai famoso (A very famous man.)
TL;DR: No, it's not vulgar.
I am going to get downvotes for this, but I feel the need to warn you.
My advice is to use assai only in written language (in sentences like "un uomo assai famoso", citing LiquidFiber, or "un locale assai noto e frequentato"), and as little as possible. If you use it, do follow LiquidFiber's advice.
In the spoken, everyday language it may appear:
The upvote-catching answer is: yes, you can use it, it's on the dictionary. But if I were a close friend of yours I would say you: avoid it! I can hardly imagine a job interview in Milan or in Florence where the candidate says assai. So embarrassing.
I think assai is fine used conversationally to mean "much/very" as described by several people above. Certainly it is used in this way in Lazio, Tuscany and Emilia-Romagna.
It has a more poetic, old-fashioned tone when used adverbially or adjectivally after the expression/word it is qualifying, which is more common in the south, esp. Naples.
In Neapolitan, it is "assaje", as in the very famous folk song "Te voglio bene assaje", referenced in Lucio Dalla's song "Caruso".
Showing you have a knowledge and awareness of this part of Italian heritage would be impressive to your Italian friends, if they are not too ignorant to know it themselves.
Assai is not a vulgar word. It is and old-fashioned word though, which was mainly used in poetry. Sometimes, is also used in southern/central Italy as part of local dialects, but even in this case, it's not vulgar at all. It is just an old-fashioned way to make a superlative (as others pointed out).