VMware Inc. is a software company that develops many suite of software products specially for providing various virtualization solutions. There are many cloud products, datacenter products, desktop products and so on.
vSphere is a software suite that comes under datacenter product. vSphere is like Microsoft Office suite which has many softwares like MS Office, MS Excel, MS Access and so on. Like Microsoft Office vsphere is also a software suite that has many softwares like vCenter, ESXi, vSphere client and so on. So the combination of all these softwares is vSphere. vSphere is not a particular software that you can install and use, “it is just a package name which contains sub components”.
ESXi and vCenter are components of vSphere. ESXi server is the most important part of vSphere. ESXi is the virtualization server. It is type 1 hypervisor. All the virtual machines or Guest OS are installed in ESXi server. To install, manage and access those virtual servers which are above of ESXi server, you will need other component of vsphere called vSphere client or vCenter. Now, vSphere client is another component of vSphere which allows administrators to connect to ESXi servers and access or manage virtual machines. vSphere client is installed on the client machine. The vSphere client is used from client machine to connect to ESXi server and perform management tasks. So now what is vCenter?
vCenter is a centralized management application that lets you manage virtual machines and ESXi hosts. It can be installed on Windows Server or Linux Server. vSphere client is used to access vCenter Server and manage the datacenter. vCenter is must for enterprises to have enterprise features like vMotion, VMware High Availability, VMware Update Manager and VMware Distributed Resource Scheduler (DRS). So vCenter is another important part of vSphere package. You have to purchase vCenter license separately.
The picture above shows in a more descriptive way. vSphere is a suite, ESXi is a hypervisor that is installed on a physical machine. vCenter is installed as virtual machine on top of ESXi server. vSphere Client is installed on laptop or desktop PC and is used to access ESXi Server and vCenter server for management purpose.