VPS stands for Virtual Private Server. Basically, it is one dedicated server split up into sections by software. Virtuozzo, manufactured by SWsoft is a popular software that can accomplish this. A VPS is one step below a dedicated server, but you gain root access to your section and it functions as if it were a dedicated server.
I'll give you a basic comparison and hopefully this will clear up the confusion. With a shared account, many websites are hosted. There is no seperation between the accounts and no account can control the server. However, with a VPS, it is split up into sections, and each VPS is independent from other VPSs on the server. Each VPS has its own processes and functions like a dedicated server--it can host websites and do whatever a dedicated server can do.