Something proper is correct or right. There's a proper fork to use for salad, and a proper way to dress for a wedding.
Things that are proper fit the situation — they're the right thing to do, because of custom or manners. A swear word is the opposite of proper language. Wearing a hat indoors isn't proper, as your grandparents have probably reminded you. Getting proper medical care means you've received the appropriate medical care. When things are proper, they're correct and suitable: they fit.