If you need more resources than shared web hosting can offer and/or full root access, both VPS and dedicated server are the solutions to consider.
When using the dedicated server you can employ 100% of the server resources. No other client will host a website on the same server. With the VPS (virtual private server), several clients are hosted on the same physical computer. However, the functionality is similar to the dedicated server. Every client has allocated some resources and the performance of his virtual server is not affected by the other customers.
VPS hosting is a more affordable way how to run your own server. You can manage it in the same way like dedicated server, you have the same root access, the only difference is in the allocated resources. If you do not need full power of the dedicated server, you can choose from many VPS hosting plans depending of your CPU, RAM and other requirements.