Hello Joshi
The terms off-premises and on-premises have to do with the ownership of the physical location of the running services.
Strictly speaking, an off-premises data center is one that is hosted on infrastructure that does not belong to the enterprise. For example, you host all of your enterprise applications on AWS or Asure. Or if you rent rack space in a physical datacenter belonging to a third party. These are examples of off-premises datacenters.
On-premises datacenters are those that are hosted at a physical location of the enterprise. Even if these exist at the headquarters for example, and they are accessed remotely by a branch office, they are still considered on premises.
An additional aspect of on-premises is that the enterprise is responsible for the hardware and infrastructure, while for off-premises, it is the hosting service (AWS, Asure, rack space provider) that is responsible for the upkeep of the physical systems. Contracts may vary as to who is responsible for what, but that’s the general idea.
I hope this has been helpful!
Laz