Many Facebook users are confused by the difference between "Pages" and "Groups," and between the new "Groups" and the old "Groups." Here's a brief-ish primer.
If you're wanting to create a Facebook presence for your church, what you want is a "Page." See my church's page at http://www.facebook.com/libertyuu for an example. To create a page, go to another page and scroll down to the bottom. On the bottom of the left column will be a link that says "Create a Page." If you click on that, it'll walk you through the steps. With pages you can post status updates which will appear in the news feed of the people who "like" (formerly "fan") the page. You can post blog posts or videos or whatever, just like with your own personal status update. You can create invents and invite all the followers to them. Pages also have the advantage of getting their own distinctive URL, which makes it easier to tell people where to find you than if your URL is something like http://www.facebook.com/?ref=home#!/pages/Unitarian-Universalist-Church-of-Arlington-VA/165807771102 (not to pick on Arlington--it was just the top of the list). If you have a long URL like that and you want to fix it, it's as easy as going to http://www.facebook.com/username/, selecting your page, and giving it a name. This is a good thing to do for branding/marketing your church. I recommend using the same name as your church's website URL and also making it the same if you're on Twitter. Our church is known in all three places by "libertyuu" -- http://www.libertyuu.org, http://www.facebook.com/libertyuu and @libertyuu on Twitter.
For ministers and other public figures, I also suggest creating a Facebook Page for yourself. Politicians and actors and musicians have these -- they're the ones you "like" and follow. Creating a Facebook Page for yourself allows you to keep your personal and professional lives a little more separate on Facebook. My page is http://www.facebook.com/revcyn, and I use it to post ministry-related links, my blog, ministry topics and questions. That way my other friends and family aren't inundated with this type of material unless they choose to follow the RevCyn page.
The old Facebook "Groups" were very similar to pages. They were so similar that it was hard to tell which an organization had or which you should choose when creating one. One advantage with these groups is that you can list the officers of a group. Another is that you can message all the group members at once. On face value, they seem more set up for churches. However, this is if you are thinking of your church's Facebook presence as for the current members only. If you're thinking of your church's Facebook presence as outreach and marketing to new potential members, then Pages are the way to go. Groups can be open, closed or secret -- you can limit who can join and who can even see that it exists. They are, therefore, more of an insider thing than a Page, which is open to everyone to see. Some examples of organizations that set themselves up as Groups are Meadville Lombard Theological School and the Church of the Larger Fellowship. Both groups still exist, but both later also created Facebook pages, which is telling about the flexibility and features of Groups for organizations like these. I'm not sure it's possible to create these sorts of larger Groups anymore. There's no "create a group" link at the bottom of their groups, like there is with pages.
Not too long ago, Facebook released a new Groups feature. With these new Groups, you can think of them as similar to Friends Lists (see my last post), but with more flexibility, and also as similar to old groups. You create these groups by going to the left-hand column on your home page, and there will be an item in the middle that says "Create a group." If there's not, click on some of the "more" options to expand your list. Once you create a group or are in a group, these groups will appear in this same column. You can post status updates in the group, and only the group members will see them, and they will appear in the news feed of members. Unless you turn off the feature, any time a member posts, you'll get a notification. In the group, click "Edit Settings" on the upper right to change these settings. What was controversial about these groups when Facebook released them is that you can put people in these groups without their permission, and then they have to opt out to get out of them, rather than opting in to get in. These groups are idea for small groups. I have one set up for my immediate family, and we post family get-togethers and family photos there. A minister might find it helpful to set up one for a committee or board. You do have to be friends with someone to add them to the group. But if you set the settings open enough, people who you are not friends with can get added, too, by another group member adding them or by finding the group through search functions, if you haven't made it a secret group. One weird thing about these groups is the status update that has been commented on most recently will appear at the top, rather than the one that was posted most recently. Also, you don't have to hit "share," just a return, to comment, making it a little more like a chat. And, speaking of chat, you can now use the Facebook chat feature to open up a chat with the whole group at once. I keep my chat "Offline," so this is a feature I haven't explored, but could come in handy.
Hope this helps with understanding the difference as you're setting up online presences, or exploring uses of Facebook in your lives. Let me know what other Facebook questions are out there, and I'll try to address them.