A Virtual Private Server (VPS) is a physical web server whose resources (such as CPU cores, RAM and hard drive space) have been allocated to multiple "virtual" servers using virtualization software. Basically, it is one physical server that has been split into multiple smaller "servers" by dividing hardware resources among clients.
The server hardware that has been allocated to a VPS are usually guaranteed to be available for use, sometimes including a higher burst rate if the server load is low and capable of providing the needed resources at the time. For example, a VPS account may come with 512MB of guaranteed RAM with a burst rate of 1GB. This means that particular VPS will always have at least 512MB of RAM available for use. However, if you experience a spike in traffic or similar event that requires more than 512MB, then extra RAM may be available (up to the burst rate of 1GB) for the situation assuming the physical server has enough unused RAM.
Unlike shared hosting, Virtual Private Servers provide root access, giving users a much higher level of control over the setup process, software installation, security and maintenance of a hosting account. As a result, the experience level required to make use of a VPS is higher than shared hosting and a unmanaged VPS is not recommended for novices or people with no experience managing a server.
Virtual private servers are independent from one another, meaning they each have their own server load, Operating System, software and file system separate from other VPS accounts. This is an advantage over shared hosting in that one client can't have a negative effect on the other Virtual Private Servers located on a physical server.