Once detached from one another, the detached parts are stubs. Generally, the word is used in the context of the half the customer retains: "When you go to the restroom, remember to bring your ticket stub with you, or you may not be allowed back in." If there's a special word for the half the box office retains, I've never heard it.
Edit, because quotations from the OED make everything better: The Oxford English Dictionary finds ticket stub in use by Ellery Queen in The Roman Hat Mystery (1929): "You'll be looking for ticket-stubs. Anything resembling half a ticket." The earliest definition given for stub, from the year 967, is as a synonym for "stump" (of a tree); many or most of the definitions that have evolved since then retain the sense of a small thing that has been severed from a larger thing—like a ticket stub.
Edit 2: Here's a reference from 1887, in a joke that does not seem to have aged very well since then:
"BATH-HOUSE ROBBER: No use lookin' fer anythin' here, Bill. Ticket stub ter one of Joe Cook's lectures, an' a poker chip. Busted drummer from Boston!" I guess you had to be there.