A Virtual Desktop Infrastructure (VDI) is a desktop-oriented service that hosts user desktop environments on remote servers and/or blade PCs. Users accessed the desktops over a network using a remote display protocol. A connection-brokering service connects users to their assigned desktop sessions. For users, this means they can access their desktop from any location, without having to use a single client device. Since the resources are centralized, users moving between work locations can still access the same desktop environment with their applications and data.[3] For IT administrators, this means a more centralized, efficient client environment that is easier to maintain and able to respond more quickly to the changing needs of the user and business.

VDS is a set of application programming interfaces (APIs) that provides a single interface for managing disks in Windows Server 2003 operating systems. VDS provides a means of managing storage hardware and disks, and for creating volumes on those disks.

This is the pooling of physical storage from multiple network devices, allowing it to appear as a single storage device, managed from a central console. It is commonly used in a SAN environment.

Virtual servers is software allowing multiple servers residing on a single server.  Most virtual servers are using VM Ware or HyperV for their software.  

A volume is an area of storage on a hard disk. A volume is formatted by using a file system, such as file allocation table (FAT) or NTFS, and typically has a drive letter assigned to it (in Windows). A single hard disk can have multiple volumes, and volumes can also span multiple disks.

Virtual Private Cloud.

To understand this, you need to understand the difference between public and private cloud. The short version is “with a private cloud you own it all and are the only person (or company) on the hardware. With public cloud, none of that is true (in most cases, but I’m not going to go into that here)”.

Disk array that has software housed within the unit and makes the unit of disks appear to the backup software as a tape library.