The terms "domain" and "domain name" are generally interchangeable. A domain name consists of numbers, letters and sometimes special characters. Domain names are easier to recall than random numbers. The characters in a domain "map" internet traffic to the assigned numbers in the IP address. If you are interested in the relationship between a domain and an IP address Wiki has a great deal of information.

For most chapters using a website builder is the best method of obtaining a domain. All of them offer the ability to purchase, renew and transfer your domain. When you purchase a domain it is not a final purchase. You are actually purchasing the right to map traffic to your domain. In order to maintain your domain ownership you will need to renew it periodically. You are not required to renew a domain and you can let it expire. You can transfer your domain or even sell your domain ownership to another entity.  

Some chapters decide it is in their best interest to purchase a domain name using a domain registrar. This option is often selected for WordPress sites, outsourced sites and sites built from scratch because it gives more control over domain ownership to the chapter. 

A source of domain registrars can be found at Hosting Facts.

I have created separate pages that go into more depth about domain registration, domain transfers and domain email.

The letters to use in your domain name is one of the most important things your chapter needs to consider. 

The process of merely "checking out" a potential domain name can escalate the purchase price on some registrar sites. Once a registrar (registration only site) knows what name you are looking for, they may put cookies into place so that when you return, the cost will escalate.

This does not happen when purchasing a domain name from website builders because purchasing a domain name from them is part of their package price. They want to gain you as a website customer so they are eager for you to acquire the name you desire.

Regardless of how your domain is purchased and registered you will need to verify your domain name ownership right with ICANN once a year. ICANN is a global organization that ensures that each internet address (domain) is unique. More information is available on their website.

REMEMBER: Domains are a totally separate product from a website even though it may be part of a website package. You can be a domain owner without being a website owner. You can be a website owner without being a domain owner. You can also have a domain registration account and not have a domain or a website.