Rivalry is the act of competing for the same thing against another person. Your rivalry with your older sister is amusing to the family but very annoying to her.
Rivalry comes from the noun rival, the person you compete against. Often a rivalry starts when people want the same reward — and their talents are just about equal. Your rivalry for the lead in the play at the local theater isn't between you and George Clooney — it's between you and the other guy who also had a great audition.