A ticket is the slip of paper or card you need to be admitted to a movie theater, a museum, or an airplane. Your ticket proves that you've paid or been invited.
Other types of tickets include the ones you get as a receipt or bill, and the kind a traffic officer hands you when you're caught driving 40 miles per hour in a 30 MPH zone. In the U.S., ticket is also used to mean "ballot," as in "I'm not excited either of the candidates on that party's ticket." If someone tells you "That's the ticket!", they mean you've made exactly the right choice.