Obscenity (from the Latin obscenus, meaning "foul, repulsive, detestable") is generally used in a legal context to describe expressions (whether words, images or actions) that offend the sexual morality of a given time and place and are not protected by the First Amendment. It also covers sexual or scatological references to the body or bodily functions (i.e. F*&k and s#$t).
Today, profanity is an expression that is specifically offensive to members of a religious group. The definition also extends to expressions that are scatological, derogatory, racist, sexist, or sexual. What is and isn't profane largely depends on the context and the company you keep.
Vulgarity (from the Latin vulgis, meaning "the common people,"), is generally coarse or crude language. For others, it is more specifically the act of substituting a coarse word in a context where a more refined expression would be expected.
View full source to read more.