As mentioned before Google Apps for Domains is a great solution for Non Profits that want to change or add e-mail to their domain.  You will need two things to get started. 

1. A registered domain name.  If you don't have that already you will want to go to Register.com or Network Solutions and register a domain name.  There are other domain registrants out there.  Many are less expensive than the two I mentioned.  However, in this case I feel you get what you pay for.  In a year or two when you need to renew the lesser competitors may not be around, or have been sold.  You then spend days trying to get access to the new system.  I have seen a lot of issues, so please use one of the registrants I mentioned.  That being said they are all gong to try and up sell you for web hosting, search engine placement, , etc.  In my opinion none of these are worth obtaining.  The only thing you want is to register the domain and have it's Domain Name Services (DNS) managed by the registrant you choose.
2. The Federal tax id number for you organization.  If don't have one yet, start filing the paperwork, but you can still register with Google.  The only downside is you will only be eligible the Standard Edition of Google Apps for Domains.  Still worth doing.  Once you do get your Federal Tax ID for you 501(c)3 you can fill out this form to upgrade your Standard Edition to the Educational Edition.

With that information we are ready to get started. 

1. Go to the sign up form.   In my opinion this form is a little confusing.   The cause of the confusion is the new Team Edition of Google Apps.  But here is magic combination.  Choose the Administrators: I own or control this domain.  I know scary, they assume you are an administrator of a system.  Don't worry about that's what where here for, to help be an e-mail administrator.  Now type in your domain name.  If you just registered it and get the "This domain is not yet registered." error don't be concerned.  It can take up to 24 hours for domain registration to be visible to the world.  Just come back to this step at a later time
   
2. Pressing get started brings us to step 2.  You just need to fill out your contact information. 
1. For number of users this should be your leadership group rounded up to the next highest 10.  So a board of 16 would probably want 20 users, may be 30.  You can always request more later.  Also don't be surprised if once you get thought the process that Google gives 200.  The are not stingy with accounts, but be respectful and only request what is reasonable for you organization.
2. Email address needs to be your currently working e-mail address.  I suggest you use one that is not within your domain name.  A Gmail Account would a great choose.
3. Don't panic on the DNS editing checkbox.  Google has great docs on how to do that.  They have step by step instruction for Register.com and Network Solutions, we can worry about that later.  Just take a breath and check the box.
4. On Type, obviously you want to check Non-Profit Organization. 
5. Now you can choose Yes, please enter Federal Tax ID.  If you don't have one, just choose the No, use Standard Edition. 
6. Size of Organization, this one is a little unclear.  In the past I put the number of current members including leadership. 
   
3. You will then need to setup your admin account.  I don't have a screen shot of the page, sorry.

Once you continue your account will be created and will be given a link to your Admin Dashboard.  You will more than likely be given the Standard Edition.  Once Google verifies your status as 501(c)3 they will upgrade you to the Educational Edition.

If you have existing e-mail accounts I would wait for the upgrade.  The additional feature included will the E-Mail migration tool.