You should register your own name, or make sure that it's renewal fees are billed only to you with Network Solutions or whoever else you risk buying your "Domain name" registration from!
Internic, a.k.a. Network Solutions was the original supplier of domain names for the net. There are also now, by new legislation, some new competitors for this service, which consists of what's called giving your website a 'nic'. Your site's name, referred to as it's domain 'nic'-name is the wordy identification by which all internet Domain Name Servers (DNS's) can "look up" your site servers electronic phone number.
While it costs money to register a business trademark, most people don't know that you can easily copyright almost anything by simply mailing yourself a registered letter containing it, that you keep stored unopened! This is legal evidence of the date and time of your discovery! You cannot legally register someone elses trademarked name but can copyright your name easily!
How DNS systems work
This is a simplified explanation of how Domain Name Servers resolve internet connections. When you type http://www.aol.com into your browsers "go to" bar, your computer first checks its own 'hosts' file and then asks your internet providers DNS server to 'look up the number' of aol.com's DNS server in it's records, which it get periodically updated from ICAAN. When it contacts AOL's DNS server, their DNS server answers back exactly what numbered server machine, on AOL's network, the web page server for aol.com (HTTP Daemon software) is currently connected on. Depending on the results your browser gets back, hopefully, the Daemon (this machines 'server' software) gives your browser the page you asked for. These basic name-number-lookup lists are updated constantly by ICAAN and copied all over the world. Unfortunately, not everybody keeps their DNS servers up to date all the time. A website that has moved to a new server may not be reachable by name, or the domain name may point to a "dead" server machine for different periods of time during changes! Your neighbor, on another internet provider, may connect to it's new server machine without any problem - because your ISP's DNS server or your computer has 'cached' some stale old DNS lookup information.
For this reason, when a site plans to move to a new host, it is best to keep both versions of the site up, both at it's old and at the new server machines for about a week, since changing every set of DNS-phonebooks (propagating new records) all over the world may take a while! These "phonebooks of phonebooks" systems connect all the valid numbered computers to the plain registered domain names (also called NICs - short for nicknames or "canonical names") that humans type and remember.
These Internet Packet(IP) address numbers consist of up to 12 digits, in 4 groups of 3 with leading zero's not included, from '1.0.0.0' to '255.255.255.255'. They resolve "geographically" by using the first, then the second, then the third etc. The first may represent a particular local (private) network or group of states, the second, an internet backbone system, the third, a particular hosting companies block of numbers, and the last, that companies certain server-machine, with your site on it. Your site usually gets it's own unique number, but not always! Normally a machine will have many last-numbers routed to it, tied to their own file folders. This page on Webdisplays is currently at http://208.235.98.25/web/dns.htm. When a domain has it's own number, it can be directly accessed by HTTP1.0 (any plain request). If a site shares the same machine number with other domain names, HTTP1.1 must be used to locate it! HTTP1.0 is the better, faster method, but the newer HTTP method requires that the server (as well as the dns-router) must also check what domain name (website) each document request is intended for. If your site is hosted with only HTTP1.1, there is no numeric request that could directly reach it. A side effect of this lame solution allows some unscrupulous HTTP1.1 site servers to feed the same website with minor automated changes as if it were dozens or hundreds of different domains, a serious spam problem.
All a website hosting company has to do is give you 2 of their DNS "phonebook" server names/numbers, a main one and a spare secondary one for their local "phonebook" machines so that you can get on with enableing (finish registering) your site at Network Solutions, or anywhere else listed at The Internet Corporation for Assigned Names and Numbers (ICAAN), that issues the domain names. To complete the setup process, after the name is rented, the registrar needs to know which 2 (or 3) DNS servers will be "resolving" your domain NIC-name to a specific server-machine at a hosting company. The technical contact at your hosting company will setup a "number" in their DNS "phonebooks" that points to your sites particular server. Only a hosting provider who uses HTTP1.0 would be able to setup a server (numbered IP address) for you *before* there is a registered domain name to be assigned to it.
Domain Name Suffixes (TLD)
You can get domain names in other suffixes also called Top Level Domains, depending on the country, from local authorities. Domains in .gov and .mil are reserved for US official use. For instance, you can get .ca domains from Canadian authorities if you live there, and some confusing, oddball suffixes, notably .cc .tv .inf and .xxx are now also available. Most registrars charge you $70 for the first two years fees and $35/yr thereafter. When you pick a name, it's often useful to get one that "speaks" to search engines! For instance, 'webdisplays' is rather useless information to a search engine. Is it about web or display or play? Web-displays.com is a better choice for those reasons - but not necessarily so for humans :-). You can rent as many domain names as you can afford however, and have them all point to the same server! You can also obtain extra visitors by buying up old, used names...
To find out if the name you want is available, you need to use a 'whois' service to check your choice against those that are already registered. Go to to Network Solutions using the little form in the sidebar to search for a name you can use! Just type in one you like to see if it's available, others will be suggested for you. A second rate TLD like .cc is of little value if it is the same name as a .com site -people will go to the .com by accident anyways, and other copyright and trademark issues will undoubtedly arise if you try to clone anothers tradename on another top level domain. Strictly speaking, the .com TLD was created for commercial as opposed to personal sites. The reason for these new TLD's is to allow the Smiths and Joneses of the world some added opportunities.
There are still many names available with the .com suffix, so chances are you can think of one that is not in use, or already on hold for your company. Once you've found a name to use, just enter your personal and billing information into the registration form and the 2 IP address numbers, and DNS server names (and email address of their technical contact person) you got from your hosting provider, and your new (empty) home will be ready to use in a few hours time! If you desire you might also park it (without the DNS or host info) free at Network Solutions or Register.com until you do find a hosting provider you are comfortable with! Never park your domain with a hosting company, use a genuine registrar or you may be sorry you let them get involved with your "free" parking!!