VPS stands for Virtual Private Server and usually refers to a single instance of a virtual machine leased for private running on a shared physical machine.
VPS stands for Virtual Private Server and usually refers to a one instance of a virtual operating system leased for private use on a shared physical machine. The physical machine typically runs a hypervisor which allows multiple of these ("virtual" or "guest") operating systems to share the same physical server.
A VPS runs its own copy of an operating system, and customers have superuser-level access to that operating system instance, so they can install almost any software that runs on that OS. For many purposes they are functionally equivalent to a dedicated physical server, and being software-defined, are able to be much more easily created and configured. They are priced much lower than an equivalent physical server, but as they share the underlying physical hardware with other VPSs, performance may be lower, and may depend on the workload of other instances on the same hardware node.