When you are proud, you are feeling pride, or satisfaction with yourself. The word proud can also mean too much of this feeling––sometimes saying someone is proud is the same as saying they're arrogant.
It's also possible to feel proud of someone else. If your best friend gets the lead in the school play, you may find yourself feeling as happy for him as if you'd got the lead yourself. When you graduate from college, your "proud parents" will want to be there to watch. The maxim "Pride comes before a fall" plays on the fact that when you are proud of what you have, you are also at risk: having something means you have something to lose.