You want to register a name server so that you can then register a domain name and have the corresponding subdomain delegated to it.
Registering name servers isn't normally done separately from registering a domain name; as part of the registration process for a domain name, you specify the domain names (and, sometimes, the IP addresses) of the name servers that will serve the corresponding zone. If the name servers you specify aren't already registered, the process will register them. See Section 1.6 for more information on that process.
If you find that you really need to register a name server independently of registering a new domain name, check your registrar's web site to see if they offer such a service. Network Solutions, for example, lets ISPs register name servers that their customers can then delegate to at https://www.netsol.com/cgi-bin/makechanges/itts/host.
Before you try to register name servers, make sure they aren't already registered. Any name server that has had even one subdomain of a top-level domain delegated to it is registered with that top-level domain's registry.
Also note that you shouldn't register a host that's not a name server, even if your registrar will let you. Some registrars (and their registries) don't check whether the host you're registering actually has any subdomains delegated to it. But if you register, say, your web server, you may have a hard time changing that information on short notice, and you may forget that your registry's name servers are giving out answers about your web server. Then, when you move your web server and change its address in your zone data, you'll wonder why some people are still trying the web server's old address.
1.7.4 See Also
Section 1.6 for registering a domain name.