If you pay admission, you're paying a fee in order to attend something, like an event or a movie. After you pay admission to the movie theater, you head straight to the popcorn stand.
The noun admission stems from the Latin word admissionem, meaning "a letting in." It often refers to a fee charged for entry, but it can also describe the process of being granted entry. The university's admission of the new student was welcome news — now he has access to the school's research facilities. An admission can also be a confession that something is true, such as your brother's surly admission that he was the one who popped your bike tire.