Someone who's skinny is thin. Your brother might be skinny enough to fit in the jeans he wore three years ago.
A skinny person is slight or lean, and your skinny jeans are tight-fitting. You might order a skinny coffee drink at a cafe, meaning you'd like it to be made with skim milk. If someone offers to give you "the skinny" on something, they're revealing some secret information. This last, informal meaning stems from military slang during World War II, probably from the idea of "naked," or undisguised, truth.