To understand what a virtual private server is, you need to understand what a virtual machine is. A virtual machine is a virtualized or emulated machine running on physical hardware. For example, people may use VMware or VirtualBox to run an additional Windows or Linux installation inside of their main operating system to gain the features of that operating system or gain isolation from their main machine by running a virtual machine.
A virtual private server is a virtual machine, but instead of running on a desktop computer, the virtual machine is running on dedicated server hardware and often uses KVM or OpenVZ for the virtualization. A hosting provider will load a dedicated server to run multiple instances of virtual private servers and sell each VPS to a client. Depending on server hardware, a single dedicated server could host anywhere from tens to hundreds of virtual servers.
In the most basic definition, a virtual private server is simply a virtual machine that is running on dedicated server hardware. When using a VPS, you get almost the same environment that you would if you were to purchase a dedicated server. You can install your own operating system and services to your server, and in most cases you are granted full root access to the server to make any required modifications to the server.
Because virtual private servers are packed into a single dedicated server, you will share resources with other VPSs on the same server. RAM is usually reserved for specific machines, allowing you to have access to the full amount of RAM that your hosting provider offered. Other resources such as CPU, I/O, and Networking are shared amount other virtual servers and can become slow if the server is hosting too many resource intensive virtual private servers.
There are many reasons to use a virtual private server, and there are cases of companies configuring virtual private servers on their dedicated hardware. The main advantage of a VPS is the price point. You can get a server you are in full control over for only a few dollars monthly. VPSs also offer the nice ability of running isolated environments which is much more secure than adding users to a chroot jail. There are many reasons why VPSs exist and the market shows that there is a large demand for affordable private servers.