The Cloud API is Meta's hosted offering. The infrastructure runs on Meta's side, which removes the cost and maintenance of self-hosting and gives faster access to new features.
It is the recommended path for almost every new project. It replaced the older On-Premise API, which Meta has been deprecating in favor of the cloud version.
Businesses can connect to the Cloud API directly or through a Business Solution Provider. The provider route adds a dashboard, automations, and support on top of the same underlying API.