A VPS is a virtual private server. It is a type of hosting plan in which you, the customer, has complete control over a virtual server. Often (and increasingly so) the VPS runs on top of cloud infrastructure (in other words — a large pool of networked computing resources, rather than a single piece of hardware).
One way to think about hosting plans is that a shared hosting plan is like renting a small apartment and a dedicated server is like a house. From this perspective, a VPS plan is like a condo.
With a VPS plan you get most of the benefits of a dedicated server, at a much lower cost. In the case of cloud-based VPS hosting, you also have the ability to scale up available resources quickly as the need increases.