A VPS, or Virtual Private Server, is a completely independent server operating system designed so it can run side by side on the same physical server as other VPSes.
Traditionally, each physical server has one operating system which is tied to that physical server's specific hardware and resources. With virtualization, the virtual server sits on top of a virtual core operating system which manages virtual resources, and that in turn speaks to the physical hardware. Meaning, to the virtual machine the physical hardware is transparent.
This has a number of advantages. Memory and processor allocation can be adjusted on the fly. A virtual machine can be easily moved to another physical server running the same core virtualization software and won't know the difference, making hardware upgrades and restoring a virtual machine when a physical piece of hardware fails much easier. Backups and restores of entire virtual machines can be done quickly and easily. And many VPSes can fit on one physical machine now, reducing overall power consumption, heat generation, cooling needs, and space requirements.
So, really the only difference is that a dedicated server is a complete physical server and one operating system devoted to you, whereas a VPS shares the server hardware and resources with other VPSes on the same server. Even though your VPS may share resources with other VPSes, because VPSes are very efficient and completely isolated your allocated resources are always available to you when you need them.
For more information on the Virtuozzo core virtualization software we use and how it is able to manage VPSes so efficiently, please visit http://www.websiteshelter.com/