An on-premise (or on-premises) system is a piece of software which is installed and run directly on the computers of the company or user in question. The whole software is thus “on the premises” of the company—on their own server and workstations.
The opposite of on-premise is off-premise, which is better known under the term cloud. In that case, the software isn’t installed locally on the computers of the company or user, but on a remote (cloud) server, to which the users can connect remotely using the internet.