Something that one does not give when the situation does not apply to them, or is uninteresting to them, or is not inherently valuable to them.
Can also be phrased as "flying-ass fuck."
Brian: "Hey Jordan, that guy is going to be at the party tonight."
Jordan: "I don't give a flying fuck, that guy is so far beneath me that worms make him eat shit off their shoes."
Brian: "Hey, did you hear that the Patriots lost the game last night?"
Jordan: "I don't give a flying-ass fuck, the Patriots can eat my cock for all I care."