If you are supposed to do something, it means that you'd better get to it. If you're supposed to go to your grandma's house for dinner, then your grandmother expects you.
Something that's required is supposed, and something that's assumed to be true — even if it's really not — is also supposed. If you describe someone as your brother's supposed best friend, it means that everyone thinks of him as a good pal, but maybe he's not as loyal as they believe him to be. If higher taxes result in supposed improvements in your city's schools, things may be better, but there's still some doubt about it.