VPS stands for Virtual Private Server. It offers almost all functionality like a Dedicated Server without assigning a dedicated hardware (Machine) to the client. VPS best suits for the websites which requires more resources than the limit set on the shared server environment but not require fully dedicated Servers. You can use a different platform (Like different OS) on VPS's.
Like Manndude said, most providers don't condone those types of activities. Unfortunately, you will find fraud and spam in any market of the hosting industry. Most hosts do a good job in combating this though.
In simple words: it is your own virtualized server with its individual operating system and hardware resources allocated from the total hardware pool available from the node (Hypervisor). They mostly operate as if they were physical servers with root access, OS re-installation, reboots, shutdown, independence, etc.
There are different levels of "VPS" out there and they are not all created equallu, and I see that people are referencing both here. You have your standard VPS and your Cloud VPS. They both are in essence the same thing, by being a Virtual Private Server. Where they differ is more at the OS level. Your standard VPS will be running on a VPS node by itself, with no failover ability, and by default the VPS will be using the Node's kernel (seeing as most people are using OpenVZ and Virtuozzo these days). With a Cloud VPS you do not have this same restriction, as a Cloud VPS is more of a server instance, meaning it uses it's own kernel. Cloud VPS's have the ability to failover from one hypervisor to another, but is not required in order to run a production enviroment.
When you are looking at a VPS you want to make sure that the provider is using a virtualization software that meshes with what you want to do. If you need FULL control of your VPS, meaning down to the kernel level then you will need to look at Cloud Services if a dedicated machine is not that you need. If you only need a VPS as a stepping stone from Shared/Reseller Hosting before moving to dedicated equipment, then almost any VPS should suite your needs.