The cloud-based software is hosted on external servers and made available over the Internet (i.e. via the cloud). Thus, the software does not have to be downloaded or installed. The whole thing is based on the "software as a service" (SaaS) principle. Specialized providers store the data on their servers. This is often paid for with a monthly or annual subscription, plus any fees for updates, support or training.
On Premise Software can be translated as "on-site software", because the software is installed and maintained locally - i.e. on the company's own servers and computers. The usage license is purchased or rented and the necessary hardware should already be available in the company.