In short — the Bitrix24 cloud is hosted on Bitrix servers and provided as a service. To register, you just need to enter your email address and invite your colleagues.
The Bitrix24 on-premise edition is hosted on your own or rented server. Maintaining the on-premise edition requires specialised skills or a contract with one of the experienced Bitrix24 partners. The main advantage of the on-premise Bitrix24 is access to the product source code and, consequently, unlimited customisation capabilities.