On-premise (often abbreviated to "on-prem") software is an umbrella term for any program housed on on-site computers of the company using the software. There's no third-party involvement with on-prem hosting, so admins are in total control over the software and IT equipment.
While it grants full ownership over assets, the on-prem model is more expensive than keeping software in the cloud or hosting at a third-party data center. On-prem hosting also entails a range of day-to-day responsibilities (equipment maintenance, troubleshooting, security, regular updates, etc.).