VPS stands for "Virtual Private Server". A VPS is one of several isolated, secure server "environments" that reside within a large and powerful physical server (think of it like an apartment within a high-rise building).
With a VPS, you get your own secure hosting environment, with the ability to install and configure whatever you desire on your server "real estate." Although there is more than one virtual server on each machine, there is no way for other users to see your VPS -- or even to know that you're there. For more information, click here.