If you ever change your web host (the actual physical location of your website files), your domain name goes with you. Your regular visitors or customers who knew your site name as (say) www.thesitewizard.com would not have to be informed about a change of web address (also known as "URL"), since as far as they are concerned, the site is still at the same place. They would simply type your domain name and they'd be transparently brought to your new location.
I’ve tried to be charitable to Verisign in my reading of Site-Finder-gate. Even then it seems quite clear that they were deep within the grey areas of their contract with ICANN, and that they knowingly acted against the spirit of the Internet that they’d been entrusted to build. That said, I respect the hustle — Verisign were, and continue to be, a publicly traded company with a responsibility to deliver value to their shareholders, and this quasi-bait-and-switch was a pretty inspired way to do it.
If you're having trouble finding a domain name (whether because of crowding or cyber-squatters), check for a help facility on each registrar's site. Domain registrars typically house search engines that return a listing of available names similar to the one you want. When you search for a domain name at Namecheap, for example, you get both the status of that name and a list of suffixes available for that name. Maybe [Sitename].com isn't available, but [Sitename].biz or .org is.
Domain names serve to identify Internet resources, such as computers, networks, and services, with a text-based label that is easier to memorize than the numerical addresses used in the Internet protocols. A domain name may represent entire collections of such resources or individual instances. Individual Internet host computers use domain names as host identifiers, also called host names. The term host name is also used for the leaf labels in the domain name system, usually without further subordinate domain name space. Host names appear as a component in Uniform Resource Locators (URLs) for Internet resources such as web sites (e.g., en.wikipedia.org).