When they're playing a film on a tv station like HBO what type of medium do they use?

Not too long ago, they were using cassettes, often Betamax. That was back when DVDs were still fairly novel. Nowadays, it’s probably a mix of cassettes, DVDs, and HDDs.
Rolls of film would only produce an image, and would need a projector behind them.

Oh, it’s complicated.

I work in commercial production at a local station, and we use MPEG2

@stratman37 But what medium? Do you just pull the mpegs from the computer, or do you use optical disks?

master control plays ‘em from a server.