Virtual Private Servers are Dedicated Servers which run special software which allows multiple users to receive a portion of the dedicated server's power, which is used to emulate the behaviour of a full dedicated server. To the user, it is as if they were running their own programs, on their own (somewhat slower) server, with full root privileges. The overhead of this subdivision is not high, and the whole virtualization process is rather efficient
Consumers of Virtual Private Servers (also known as VPS) typically require more bandwidth, processing power and/or have more traffic than users of Shared Hosting accounts. They might also require a virtual private server because they run special software which is unavailable in a shared hosting environment or requires a privileged account to run. The big players in the web world, with large traffic and processing needs, will lease dedicated servers which will run for them exclusively. As computers have become more powerful, a subdivided high-end dedicated server, running VPS software, can provide a user with a virtual server which is as powerful as physical dedicated servers were a few years ago.