A Virtual Private Server (alias VPS) is a server environement simulated by a special virtualization software that assigns to it a part of the resources of the physical server (or physical node) that hosts the VPS such as memory, CPU and disk space.
A physical node can host multiple VPS, each with a different amount of resources assigned that can also be dynamically changed over time, while a distinct installation of different Operating Systems can run on each VPS, completely isolated from the ones installed on the other VPS, just like if they were distinct physical servers.
By hosting multiple VPS on the same physical node VPS users can share the costs of the whole physical node thus saving money but not renouncing to the advantages usually available only by having a dedicated server at their disposal, here is a brief summary of the advantages of the VPS offers:

• the security resulting by isolating your own web sites, data and software from the other people's ones
• the warranty to have at your exclusive disposal all of the assigned server resources all the time and not suffering of any performance glitch even if web sites or applications hosted in other VPS on the same physical server used all the server resources at their disposal
• the convenience to benefit of features typically available on enterprise class servers (hot plug and redundant disks, power supplies, fans and other hardware parts, multiple CPUs, superior overall quality and reliability, etc.) at a cheaper cost than an entry level server
• the ability to customize your system configuration to match your own requirements, free from other users' constraints