In years past sub-domain names were more readily found since server space was left to the pros with free site space being attractive and domain forwarding not as accessible to an average non-tech user. You would have your page name, the server name and the extension; usually .com at that time.
A standard domain might look similar to: www.mydomain.com your web server would go into the mydomain area and your extension could be something other than .com (.net, .org. edu to name a few).
Today you might find sub-domains used for the free blogs like blogger.com or on a larger scale you could have a company site that has sub-domain pages for various departments.
A sub-domain might look similar to: mypage.mydomain.com your specific page would go into the mypage field is the only difference from the domain.
A pro to using the sub-domain is if the server name is a large company that is well marketed your page will might come up sooner on searches by having that association. What I personally do is keep the sub-domain name, and forward an easy to remember domain name of my own.
This way I have the benefit of the association with the larger server name as well as an easy to remember name that will take my visitors directly to my page.