Real is an adjective. It only modifies nouns or pronouns.
Really is an adverb. It modifies verbs, adjectives, or other adverbs.
Correct: He stayed at hotels with real class.
(Class is a noun. The adjective modifies it.)
Incorrect: He stayed at a real classy hotel.(Classy is an adjective. It should be modified by an adverb.)
Correct: He stayed at a really classy hotel.
(The adjective classy is modified by the adverb.)
See also Sure or Surely?