It means that you are not interested in or worried about something or someone; that you place no value on something, an event or a person; that you do not care and are uninterested. It means that you place less value on it than you do on a turd, says it all really.
“Malcolm had a hissy fit when he found out he wasn’t invited to your birthday bash.”
“I don’t give a shit.”
When you really really want to emphasize that no one cares about something. Not only does it feel satisfying to say, it's a quick and easy way to shut someone down when they say something stupid or trivial.
Guy 1: My favorite color is white.
Guy 2: No one gives a shit.
This is a reply that you give to an announcement that in inconsequential, especially when the person making the announcement thinks it's some sort of big deal, or even notable.
You won a Golden Globe for that? Yes, I did. Wow...file that under Who Gives a Shit