cwtnospam has explained it already, but here's an analogy to make it more clear:

Suppose that the residents of your neighbourhood decided on a name for your area (e.g. HogPatch). That works fine locally, but if you want anybody from the outside to be able to mail you a letter, you need the name to be known by the nationwide post office.

I already answered this above - see the section on "dynamic DNS".
You could also just tell your correspondents to use URLs involving the IP address of your router instead of a domain name.

I assume you mean a web server. A web server can be quite useful just within the local network. And if you arrange a domain name (see above) and also configure the "port forwarding" (search these forums or just google) then you can have a globally accessible web server.

These companies pay for the right to be a registrar - someone who registers domain names. Ultimately it comes down to ICANN: http://www.icann.org/

Just a question..

Do you have a dynamic outgoing ip?

Just asking because that's what I have but there's this free dns service called
dyndns - http://www.dyndns.com/services/dns/dyndns/

It kinda sounds like you want something like that...
also what you would need then is this app:
dns updater (I think you can download it from the same site as well).

Basically all you need to do is register on the site, choose a name for you site
www.yourname.xxx.com
(here you can find the different choices for the ending of your url -
http://www.dyndns.com/services/dns/dyndns/domains.html )

After choosing the name you'll have to give them your outgoing ip and stuff like that, then you might have to forward port 80 to the internal ip of the computer running the webserver. Then download dns updater and configure it (pretty straight forward) and then see the magic happen =))

Not sure if you meant something like this, but it's fun playing around with anyway =))

laterz
dav.