Real is an adjective that describes a noun (a real diamond, real events), while really is an adverb that describes a verb or adjective (really appreciate, really long). In careful writing, "that's real nice" should be "that's really nice," since nice is an adjective being intensified, not a noun being described.
The core difference
Real describes nouns as genuine or actually existing; really describes verbs and adjectives, meaning "truly" or "very."
- real — actually existing or genuine (adjective): The story is based on real events.
- really — in actual fact; truly, or used to mean "very" (adverb): That was a really long flight.
How to tell them apart
The mix-up mirrors good and well: real is an adjective, so it belongs in front of a noun — a real diamond, real events, real concern. Really is an adverb, so it belongs in front of a verb or another adjective — she really did finish the marathon, a really long flight. Informally, real is sometimes used as an intensifier too, as in "that's real nice of you," but this is a casual usage rather than the standard form, which is really nice.
Really also has a separate life as a one-word question or exclamation — "Really?" — expressing surprise or asking for confirmation, a use real doesn't share. If you're unsure which to pick, check what follows: a noun wants real, an adjective or verb wants really.