Use of gerunds after subjects

In the statement, "A man sitting next to you at a movie theater was very disruptive," what is "sitting" - a gerund, a verb, or an adjective?

"A man sitting next to you" itself is a noun phrase. The word "man" is the head noun, with 'sitting next to you' functioning as its post-modifier. In terms of grammatical form, 'sitting next to you' is a (non-finite) present participle clause. So, 'sitting' is a verb, and 'next to you' is a prepositional phrase functioning as place adverbial. The key to the question is to distinguish between two easily confused concepts 'form' and 'function'.

"Sitting" in this case is a present participle used as an adjective.

See an explanation at Medium.

You might have learned that a verb with the ending -ing is sometimes called a gerund, and gerunds can be used as both nouns and adjectives. Present participles of verbs are formed the same way, as in "I am sitting".

The site I referred to above explains that "sitting" would be a gerund used as an adjective if you were talking about a "sitting room", because the room is not doing the sitting. It is for people who want to sit somewhere. "Sitting" in your sentence is a present participle used as an adjective, because the man (the noun that "sitting" and the clause it belongs to modify) is doing the sitting himself.