A Virtual Private Server (VPS) is a virtualized server. A VPS hosting environment mimics a dedicated server within a shared hosting environment. It is technically both shared hosting and dedicated hosting.
With dedicated servers, you rent an entire server. This is optimal for people that have very high traffic to their websites or need to set up their server in a very specific way. Not everyone needs to have a fully dedicated web server, however. If you're just getting started with your website, you can save quite a bit of money if you rent a small portion of the server. Shared hosting is when you share a portion of the server with other users rather than rent an entire server to yourself. The technology behind VPS hosting is similar to that of VMware or Virtual Box. These programs allow you to run several virtualized operating systems on one machine. For example, your desktop may be running Windows 7, but you can also run other operating systems such as Windows XP or Linux without needing to restart your computer.
VPS hosting environments function in the same manner. While you have one physical server, you can run several different virtualized operating systems on that one server. Each virtualized operating system acts as if it was a dedicated server. This allows you to have all of the benefits of a dedicated server at a fraction of the cost.