The user gets his or her own Web server but is not allowed full control over it (user is denied root access for Linux/administrator access for Windows); however, they are allowed to manage their data via FTP or other remote management tools. The user is disallowed full control so that the provider can guarantee quality of service by not allowing the user to modify the server or potentially create configuration problems. The user typically does not own the server. The server is leased to the client.
By definition, a domain name is simply a human readable form of an IP address. In function it is the destination that you type into a web browser in order to visit a website, such a www.google.com. Metaphorically, it is very similar to how you would scroll to a contact in your cell phone rather than manually dialing the person by entering their full phone number; the phone number would be an IP address and the saved contact would be a domain name. Always buy a domain with a reputable domain registrar.
On the topic of dedicated hosting, many web hosting services also offer managed hosting. This type of hosting sees the web host act as your IT department, handling a server's maintenance and upkeep. This hosting option is something that you'd typically find with dedicated servers, so it's a business-centric addition. Naturally, it adds a few bucks to the hosting cost, but nothing that should break the bank if you have the resources for a dedicated server.