VPS stands for Virtual Private Server.
Typically when you buy hosting you will be paying for shared hosting. In other words, there will be hundreds of other sites on the same server sharing all the servers resources.
As a site grows and gets busier it will use more and more resources. On a shared server this can be an issue if one site is slowing down the server for the other sites. Your host may ask you to upgrade to a VPS.
A VPS is basically a dedicated server split into smaller segments giving each user guaranteed resources for their use.
There may just be three or four people sharing a single server giving them all more resources. This, of course, will be more expensive than a traditional hosting account as you are getting more space and more resources.