Leave it. Google is smart enough to know that www and non-www points to the same content. Unconcern yourself with this. No redirects are going on here, don't bother putting any in place (unless you need to for other technical reasons, which do come up from time to time).
The only issue I have with this comment is that Google views www.website.com and website.com as a seperate page. If you would ever like to get a good page rank, it needs to be only one.
You simply need to do what you read about. redirect one to the other. This way if you want it to be with the "www." it will show the "www." even if someone doesn't put it in.
For example. type "google.com" into your search bar. what happens?
Furthermore if you do not redirect it, you will have to register both forms of the domain name in Google's Webmaster Tools.
Now about your initial question, I agree with aj_nsc first post. The domain name is the address and the server files are the website at that address.
I could explain how a Domain Name Works, but I think this Excerpt from Wikipedia does the job.
Here is how a Domain Name Works:
Domain name syntax
A domain name consists of one or more parts, technically called labels, that are conventionally concatenated, and delimited by dots, such as example.com.
[*]The right-most label conveys the top-level domain; for example, the domain name www.example.com belongs to the top-level domain com.
[*]The hierarchy of domains descends from the right to the left label in the name; each label to the left specifies a subdivision, or subdomain of the domain to the right. For example: the label "example" specifies a subdomain of the "com" domain, and "www" is a subdomain of "example.com". This tree of labels may consist of 127 levels. Each label may contain up to 63 ASCII characters. The full domain name may not exceed a total length of 253 characters. In practice, some domain registries may have shorter limits.
[*]A hostname is a domain name that has at least one associated IP address. For example, the domain names www.example.com and example.com are also hostnames, whereas the com domain is not. However, other top-level domains, particularly country code top-level domains, may indeed have an IP address, and if so, they are also hostnames.