VPS stands for Virtual Private Server. As the name implies, it is a Virtual server, unlike the dedicated servers which cost far more to lease.
You can think of a VPS as an independent slice of the dedicated server it is running on. The slices cannot affect one another, and you have total control over how you setup your VPS. You get the flexibility of a dedicated server while keeping costs low.
The reason most people here use a VPS is either because their PCs aren't upto the task, or they cannot run their PCs 24/7.
Hope that helps.