In short - Bitrix24 Cloud is hosted on Bitrix24's servers and provided as a service. To register, you only need an email address — then you can invite your colleagues and start working.
Bitrix24 On-Premise is hosted on your own server or a rented one. Maintaining it requires specialized skills or a contract with an experienced Bitrix24 partner.
The main advantage of On-Premise is access to the source code, which gives you unlimited customization capabilities.