1) VMware vSphere ESXi is the hypervisor you need to install on a server (you can download it for free)
2) VMware vSphere is the name of the suite that contains a lot of components (example: ESXi) and features that you can use to take advantage of from the ESXi hosts. This is based on the vSphere license/edition you purchase. You can see what each edition features: http://www.vmware.com/vmwarestore/vsphere_purchaseoptions.html
3) VMware vCenter is a Management tool to manage multiple vSphere ESXi hosts and virtual machines
4) vSphere Client is a client tool that you use to connect to your ESXi server or to a vCenter server to manage ESXi host/s and Virtual Machines
If you want to install ESXi and use its basic features, you do not have to purchase the license. You can just use the free edition.
EDIT: vSphere Web Client is a new component featured in vSphere 5.0 that lets you manage ESXi hosts and virtual machines. It is a light-weight component. This is basically a stripped down version of vSphere Client, that runs over the web.