www is the commonly accepted practice for designating that specific URL's serves up web traffic, there is no rule that says your URL have to begin with
Domains consist of a few parts:
The trailing dot is intentional, all addresses are understood to end with a period but technology lets us omit it for convenience.
com. - Top Level Domain
example. - Your domain
image. - Sub domain
www. - Sub domain
- and so on...
As an analogy, think of it as a folder on your computer for example, but having it written it down in the opposite direction.
Once you have access to the folder
example, you can do what ever you want with any/all folders inside it (regardless of how deep they are) but you don't have access to
com nor the root drive.
When you register for a specific domain you typically have full control over all sub domains. If a registrar ever makes you pay for a sub domain I'd immediately switch to some one else. In this case if you owned
example.com you could do what ever you want with its subdomains. Heck you could do
this.is.my.awesome.site.example.com if you wanted.
That being said, fewer subdomains is generally better when it comes to people having to actually type them in. Most places would have both
www.example.com point to the same site, and just include them both options to cover all their bases. But nothing is stopping you from making your domain names whatever you want. If you're using an online provider like GoDaddy or the like, they'll likely have an built in DNS management module that will let you configure your own DNS records. You'll have to add the subdomain there first, then you'll need to configure your website to respond to requests made for that specific URL. Typically referred to as the bindings. Once those two parts are done you should be able to hit the site using your new URL.