Someone incorrigible seems to be beyond correcting, improving, or changing. When you talk about an incorrigible bully, you're saying they're always going to push other people around.
Even though incorrigible implies a person is kind of hopeless, it's often used as a light-hearted word. When we describe someone as an incorrigible flirt or as an incorrigible gossip, we mean that this is just the way they are, and it would be foolish to try and change them. If someone is just being obnoxious as usual, you could say "You're incorrigible!"
adj impervious to correction by punishment
unrepentant and incapable of being reformed
uncontrollable, uncorrectable, unmanageable
incapable of being controlled or managed
not obeying or complying with commands of those in authority