There are two ways to acquire ESXi. As a free download suggested above and as an Embedded Hypervisor. The advantage of using an Embedded Hypervisor is you get to have real-time backups via the vMA appliance. The disadvantage of the Embedded Hypervisor is you can only buy it with the purchase of a new server.
The VSphere Client is the host management software that comes with ESXi and ESX.