While I am a newbie when it comes to VMware, but I chanced upon this question, while looking for an answer, so consider the answer as (not yet) not authoritative.
My understanding is that "vSphere" is a brand name for the suite of software products, that allow an entity (enterprise, hosting co. etc.) to create cloud-computing infrastructure, using virtualization technologies, but at the same time giving tools to configure and manage cluster of hosts running hypervisor and VMs (on the hypervisor).
Also, my understanding is vSphere Hypervisor includes ESXi for the server virtualization, but bundled with just enough management tools to manage a single instance of ESXi, i.e. it is not cluster-aware, and can manage only 1 host. That single host, however can run many VMs.
The suite, is extended by additional optional (& licensed) products that take care of functionality s.a. Application Discovery, Data Protection, Replication, Security.