This article explains briefly my understanding of a Virtual Private Server (VPS), in contrast to a dedicated server and a shared server.
Some years ago a typical server was a dedicated server, i.e. a hardware box you own and you use on your own premises. You just installed the relevant software for your server and the server provided the enabled services to all connected users. Nowadays servers (I mean the hardware box) are often provided in large data centres. These data centres may have thousands of servers installed and may rent their servers to thousands of individual customers. As long as you have the access to your rented server and not sharing it with other data centre customers, this can be understood as a dedicated server.
If a server, be it in a data centre or on your own premises, is dedicated just to one customer, this customer has to cover all running costs of the server. And with today's hardware performance the capability of one server may not be used efficiently. A modern server can handle hundreds or even thousands of low performance web sites, for example. This is where the concept of a shared server may come in: a server is shared between thousands of individual customers. A typical web hoster host your web site on hard- and software that is shared with many other customers. Normally you can only access you private area and do not have full access to the server. What you can do with the server is limited by the set-up defined and authorization given by the web hoster.
The virtual private server (VPS) goes a step further: you share the server (hardware box) with many other customers, but your area is running in a virtual machine, which can be understood as an emulated computer system. Although the hardware is shared, the look and feel is as if you are using your own server. A VPS runs its own copy of an operating system (OS), and customers may 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. However, as they share the underlying physical hardware with other VPS's, performance may be lower, depending on the workload of any other executing virtual machines.