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.
The physical server typically runs a hypervisor which is tasked with creating, releasing, and managing the resources of "guest" operating systems, or virtual machines. These guest operating systems are allocated a share of resources of the physical server, typically in a manner in which the guest is not aware of any other physical resources save for those allocated to it by the hypervisor. As a VPS runs its own copy of its operating system, customers have superuser-level access to that operating system instance, and can install almost any software that runs on the OS, however due to the number of virtualization clients typically running on a single machine, a VPS generally has limited processor time, RAM, and disk space.
Although VMware and Hyper-V dominate in-house corporate virtualization, they are less common for VPS providers, mainly due to cost and limitations - who typically use products such as OpenVZ, Virtuozzo, Xen or KVM.