VPS is the acronym for
Virtual Private Server. Hosting companies provide customers the ability to rent computing resources, which are used to create and run virtual machines that are controlled by the customer, but run on the hosting company's hardware. Such a VM is known as a VPS.
Typically, companies that rent VPSes have large VM "farms" where many customers and clients will run VMs that share the same physical host(s), but are otherwise completely independent from one another. These physical host servers are divided up to create many virtual servers, using one of the many virtualization hypervisors on the market. Each VPS can run its own full-fledged operating system and can be independently operated and rebooted.
In the world of commercial web hosting, VPSes exists between shared and dedicated hosting.
The concept of VPS may also be known by the terms "Virtual Server" and "Virtual Dedicated Server."
Wikipedia has a page on Virtual Private Servers that goes more in depth into the concept of virtualization and some of the distinctions between different types of VPS offerings.