"How again was it that you convinced me to do this?" the boy said to his friend before they jumped out of the plane on their first sky dive. Convince means to talk someone into something, or win someone over.
A person convinces by making an argument—you can convince your parents to let you go to the movies by promising you'll do all your homework the next day. A thing can convince as well. The cash prize convinced me to enter the contest. The threat of failure convinced the kids to study.