Domain-Name: 
Every server on the internet has a unique number, the IP address. As these IP addresses are not that easy to remember, there is a system assigning names to all these IP numbers.
These names all end with a dot and a 2 or 3 chars long, so called, top-level-domain. There are various organisations 'owning' these top-level-domains (TLD). The 2 chars long TLDs are country codes, the 3 chars long TLDs (.net, .com, .org) are controlled by InterNic.
The name in front of this dot and the TLD is your choice. It has to be registered with the organisation 'owning' the TLD behind it and it has to be unique in that TLD. Please check out the search page to find a free name.
Before this domain-name most http-servers carry a 'www' as sub-domain name. But we can also add more and other names in front of your domain-name, maybe to run different sites for one company, all in the same domain. 
In general it looks something like 'mychoiceornothing.mychoice.ihavetoregisterthis.tld'.  
Importance: 
The domain-name will show up in the address of your web page, like 'http://www.mydomain.tld/', and in your e-mail address, like 'myself@mydomain.tld'.
Most times this information will be transferred and submitted without anyone´s notice, so it isn't that important. But sometimes you or others may have to type it, so it's real important that it is somehow short.
As today the most average and popular names are allready in use, it may take some time to find a free domain-name which meets your demands. The best chance to find a free name is to add a dash, maybe with your location, like 'mycompany-NY.com'. See the next page to check for available domain names.
If you don't want to waste your time, searching for a name, simply ask us to suggest some names. There are many unused names, we surely find a solution.