The user gets his or her own Web server and gains full control over it (user has root access for Linux/administrator access for Windows); however, the user typically does not own the server. One type of dedicated hosting is self-managed or unmanaged. This is usually the least expensive for dedicated plans. The user has full administrative access to the server, which means the client is responsible for the security and maintenance of his own dedicated server.
You could think of the sites that share your server as your roommates; there's really not that much separating you from them. Sure, you can close the bedroom door, but they can still cause nightmares for you in the kitchen and the bathroom. In web hosting terms, all the sites share a single server's resources, so huge traffic spike on Site A may impact the neighboring sites' performances. It's even possible that another site could take down the shared server altogether, if it crashed hard enough.
Usually a single machine placed in a private residence can be used to host one or more web sites from a usually consumer-grade broadband connection. These can be purpose-built machines or more commonly old PCs. Some ISPs actively attempt to block home servers by disallowing incoming requests to TCP port 80 of the user's connection and by refusing to provide static IP addresses. A common way to attain a reliable DNS host name is by creating an account with a dynamic DNS service. A dynamic DNS service will automatically change the IP address that a URL points to when the IP address changes.