Virtual Private Servers (VPS) can be an appealing alternative to the dedicated servers. Whereas dedicated servers can be relatively expensive the VPS gives a lot of the advantages that are offered from a dedicated server but at a lower price. 

The reason why a VPS is more affordable is that the server is physically still shared with others, but it's typically done so that each virtual server can run its own operating system, and each server can be independently rebooted. Furthermore, you will also have root-access on a VPS, which is usually not allowed in reseller hosting or shared hosting. 

Each virtual machine, as a VPS can be referred to, operates in principle as a dedicated server, but in reality you do share with others. If the hosting company has not pushed the limits (system resources) on how many VPS accounts they set up on one server, the VPS will run like a dedicated server. 

The fact that the server is only "virtually" private has one extra advantage. If you are not a specialist and unsure if you have the skills to run a fully dedicated server, a VPS is a good starting point. The thing is you are not 100% on your own. Good and reputable hosts offering virtual private servers with will typically be available for support in case you mess up something in your settings, or the server will will not surface after a re-boot, etc. Having the hosting company service and maintain the server isn't a bad thing.  

Other names used by hosts for Virtual Privte Servers are "Virtual Dedicated Servers" or "Dynamic Dedicated Servers".