Something that's off-color is offensive or vulgar. Your parents might stop inviting your uncle for Thanksgiving after years of his embarrassing, off-color jokes.
You'll most often find the adjective off-color describing jokes that make some listeners laugh, but offend or disgust others. Standup comedians are frequently famous for their off-color humor, while politicians caught telling off-color stories can face serious career difficulties. Before American English adopted off-color to mean "in questionable taste," this adjective was used to describe faulty gemstones.