i realised this on windows system but the method must be the same on unix

You must buy a domain name from an organism ( ovh for exemple )
After the registeration accepted you must do monitored you dns settings ( from organism ) to point your dns on your static IP

ex : www.toto.com ---> 148.114.145.111
look on the web the differents organism who permits to buy a domain name

Generaly after you have access a forum to help to manage you domain name ( configure mx redirection, cname, .... )
good luck

P.S : you talk about a public domain or a private domain ??