Definition
A VPS (virtual private server) is a virtual machine carved out of a physical server and dedicated to a single user: it has its own operating system, a defined quota of CPU, memory and disk, and full administrative access.
How it works
A provider runs multiple virtual machines on one physical server using a hypervisor, keeping them isolated from one another; what the user gets is a “virtual machine” on which they can install software and reboot at will, usually billed monthly or hourly.
Usage
A VPS suits hosting websites, running an app server and databases, setting up proxies, or development and testing: cheaper than a physical server and more flexible than shared hosting. Administrators usually manage a VPS by logging in over SSH.
Related and confusable terms
A VPS is essentially a virtual machine rented to you; it is one selling form of a server, distinct from a dedicated server where the whole machine is yours. Large elastic pools of resources belong to cloud computing instead.