Proud is a general word to describe somebody who is pleased with himself, or with what he has achieved. Arrogant is a disapproving word to describe somebody who thinks he’s better than other people. Conceited is a disapproving word to describe somebody who thinks he or his achievements are better than they really are.
proud His parents are proud of him. She’s proud of being in the team.
arrogant He’s first in his class, but he’s not arrogant about it. She’s too arrogant to learn from her own errors.
conceited I knew I was going to win, and I’m not being conceited. That conceited singer didn’t want to sign autographs to the people waiting outside.