VPS: A Virtual Private Server, still involves hosting several websites on the same server, but there's a hypervisor that ensures that one website doesn't start causing trouble for the other sites on the server. With VPS Hosting, you're allocated a certain amount of RAM, bandwidth and server space.
Dedicated: You're able to choose the equipment, including the amount and type of RAM, the size and type of hard drive and the processor, used in the server. Further, 100 percent of the server's resources are allocated to your site and they aren't shared with any others.