A domain name, as many will know is what you type into a browser to access a website. Without a domain name, everyone accessing your server/website would need to be entering the IP address.
Choose a registrar
First thing is to choose a registrar (who you are leasing the domain from). You can search for "domain name registrars" and find who is cheapest. So long as they handle DNS (which most do) you're good.
I'm currently using tsohost.com, as they're pretty cheap, and besides a few little issues, it works for me.
Choose a domain name
On the registrar's website there will be a section to purchase a domain. Upon clicking this you'll likely be greeted with a searchbar, search for whatever domain you'd like here, and they'll let you know if it's available, and what similar domains there are.
Select the domain(s) you wish, and add it/them to your cart.
Purchase your domain name
Simply checkout, and make your way through the process. Tada, you now have a domain name!
Next up you'll need to link your server and domain name with DNS.