Non Profit Tech blog has posted an excellent start to their 3 part series on How to get your small nonprofit up on the Web, showing how to register for your domain name and why you want one. I'm also heartened to see that part 2 will detail how to use Google Apps for Domains as your email hosting provider, for two reasons. First, we're in the process of switching our email infrastructure from hosting it ourselves to GAD - it made to much sense from a cost, reliability, and support angle. Second, this past weekend I spent time moving my personal email to GAD for the same reason, and found the whole process to be well documented and structured by Google. At every step of the way the instructions are pretty clear and not overly technical. Where it does veer into technical details, there are specific, step by step guides to follow.
One caution about registering domain names - make sure that the email contact address you have for both the Administrative and Technical contacts is a reliable email address that you check regularly. You'll want to keep an eye out for bogus DNS transfers, and more importantly, when your domain name comes up to renewal you'll be notified there. You absolutely do not want your domain to expire, since another person could quickly come along and register it.
Once you have your domain name, you'll likely need DNS hosting. DNS hosting lets route host names, mail, and other services to servers by associating a numerical IP address with, you can read up on DNS hosting services on wikipedia. If you register with Godaddy, I believe they offer that as part of your registration, as most registars do. You may not want to use a registrar's DNS hosting to insulate yourself from having to also move DNS hosting if you switch registrars in the future. Using a separate DNS hosting provider can prevent downtime for your web site and services int hat case. There are many free DNS hosting services available, one to look at is everydns.net, which is free but accepts donations.