VPS
Also known as: virtual private server
A VPS is a virtual machine rented from a provider, with its own allocated processor, memory and storage, and full administrative access. Several are hosted on one physical machine, but each behaves as an independent server.
It occupies the middle ground between shared hosting, where the environment is fixed and neighbours can affect you, and a dedicated machine. You install what you like and configure it as you wish, which is the point.
The trade is responsibility. Operating system updates, firewall, backups and monitoring are yours unless you buy them as a managed service, and a VPS that nobody patches is a liability rather than an asset.