You dont need a domain name actually at all. Depending on what you are doing, if you have a static public address, you can use the ip to access the server. This is fine if the server is being used for extranet, backups etc. But if you have a public face, then youll need a domain name. At present nz domains can be had for around $25 a year, and .coms for $15.
Dot coms are a pretty congested namespace, and you will have to think pretty hard to find a short name that isnt taken. Sit yourself down with a thesaurus, and either on online domain name checker or a command line whois. Some domains that are taken, are actually being held for on sale, and so might still be available for purchase at a premium. In that case typing the domain name into a browser will produce some contact information. The whole process works rather like personalised number plates. Theres an official outlet as well as private resales.
Transfers
If you have a server running and want to transfer a domain name to it then theres things to remember: