Does a DNS server allow users to make their own domain names?

Me and my friends want to build our own server to make a web hosting business. We plan on installing WHM and Cpanel aswell as a DNS server.

We want users to be able to have their own webspace aswell as give them the ability to choose their own domain name (www.theirdomainname.com)

just the DNS server will not be enough. You also need to Register a domain name from a Registrar and then associate your DNS servers (you need at least two DNS servers, possibly located in two different networks in order to provide redundancy) to the domain you just bought.

Accordingly to the extension (ex. .com or .it) there is a different entity, also called Sponsoring Organization, responsible for the top level domain. You can find a complete list of Sponsoring Organization responsible for domain name registration here:

http://www.iana.org/domains/root/db/

In general the Sponsoring Organization is not accepting domain name registration directly, they instead accredit Registrars and you have to register the domain through one of them.

Some of the most know Registrars for top level domains (ex. .com/.net/.org/.net/.biz/.info) are listed here:

http://www.register.com
http://www.internet.bs
http://www.enom.com
http://www.namebay.com
http://www.sibername.com

Cpanel is not offering registration services by itself, your hosting company (or yourself) has configured it to register domains using one or more Registrars, they (or you) do it using Cpanel modules and customizations, as a result when you ask to register a domain via Cpanel you are basically executing a special script configured by your hosting company (or yourself) and the script will deal with one or more Registrars.