VMWare sphere is the name for an application suite developed by VMWare for the purpose of managing virtual machines. It is the collection of ESXi, vCenter, and vSphere client that makes up vSphere.
vSphere client = An application you can install inside an OS to view and manage ESXI VMs. You simply type the hostname and root password for the ESXI host to jump over to it.
vCenter = Is an enterprise application software that you can install in a Windows VM on top of a ESXI hypervisor in order to manage multiple VMs. You have to have a specific license for this.
- ESXI can be explained in ESXI but is essentially a hypervisor VM that is the actual OS.