The adjective on-premises describes something that is on a particular property. If a company has an on-premises computer server, the server is located on the company's own property, not run by a third party somewhere else.
On-premises is usually used before the noun it describes, as in the example above. A hotel may have an on-premises hair salon, where guests can get a haircut right at the hotel. It may also have an on-premises restaurant for getting a bite to eat without having to go out. On-premises is often used in the realm of information technology (IT): For example, "on-premises software" runs on a company's own computer network, as opposed to being generally accessible through the internet.