Check-out is another term for checking out (not only) of accommodation services. It usually takes place at the reception desk, where the receptionist will check you out, take your card or room key, and check your payment method. Alternatively, you will pay for your accommodation with her and receive proof of payment.
Some modern hotels offer more than one way to check out. Sometimes you just need to drop your card or room key into a locked box, and the receptionist will then check out the guests at the same time after collecting all the cards/keys from the box. Some hotels allow you to check-out using a check-in app (e.g. Alfred, MyStay, etc.).
The opposite of check-out is check-in.