VPS is Virtual Private Server. It is where you are on a shared server but without all the other people there, sort of like an intimate dinner party for just 10 people. You get SSh access and it feels like a dedicated server for the most part but you do still share the box with a few others.
Also known as Virtual Dedicated Server(VDS)... It's the missing link between a shared server and a full dedicated server. It is just a server whereby you get a portion of the RAM, HD and Bandwidth etc, without interferring others on that same server. As it involves partitioning, you'll feel feels it's like a "mini" dedicated server with root access to it.