Facebook does not provide a way to have groups that anyone can join instantly. Someone already in the group must add them, or they can request approval from a group admin.
The likely reason for not providing this is that it would attract spammers. Old-style open groups had spam problems and they did not even have the new group notification or chat options. Unlike posting to a page, all group members are now notified by default of any new post to a group, and also for groups with no more than 250 members it is possible to send chat messages to all members of the group including people who are not confirmed friends.
If you want anyone to be able to join and don't care about the notification or group chat features, then you may want to use a page rather than a group.
I use the same type of group for my Fraternity's page on Facebook. We need to let members know information that is going on, and some of which can't be seen by others (especially uninitiated members :D).
The following is a screenshot a setting under "Edit Group" (you can get to this by going to the group page, then clicking on the button at the top right of your page).
Click here to see.
What you need to do is either have a "Closed" or "Secret" group (which I currently have my group set at).
Open Group
"Anyone can see the group, who's in it, and what members post."
This is typically used by businesses and things of that nature that want to get the word out that, for example, there will be a sale on Saturday or a special on beer. Anyone can see it and there are no restrictions.
Closed Group
"Anyone can see the group and who's in it. Only members see posts."
This can be used by a business or an organization like what my fraternity is. Basically you are giving the people who are in the group access to the posts. BUT, different from "Secret" group, people can actually see the page in search (and can ask to be in the group).
Secret Group
"Only members see the group, who's in it, and what members post."
As mentioned above, this type of page does NOT show up in search because the administrator only wants those people in the group to be able to see the posts. This goes great for a fraternity, because certain things such as ritual should be secret to only those initiated.
Basically how I see this, if you only want people in the group to see posts, you have the option of "Closed" or "Secret". From there you can answer the questions:
(1) Do I want the group to show up in search?
(2) Do I want people to see the group from the outside, and be able to request in?
Hope this is helpful... if you have any other questions about groups you can go to facebook and they have many help and FAQ sections to help (I'm limited to two links in this post because I'm new).
Best Answer
You must have a new-style group. Only old-style open groups (which cannot be created any more) can be joined without approval.
Previously, open Groups were similar to Pages and anyone could join. They were so similar that a lot of people were confused as to which they should use, and they would often complain because they created the wrong kind and would want to switch (which is not possible). Facebook redesigned Groups in October 2010 to make them better for small groups of friends, with features such as group chat and new notification options. Any member can add new members from their own friends list, or others can request to join with group admin approval. Pages on the other hand are designed for large numbers of people, and for any case where you want anyone to be able to join without approval.
Although each group member can change their own settings, by default all group members are notified when there is any new content from any member in one of their new-style groups. This is very helpful for small groups of friends, however it does not really work if anyone can join. The reason is that spammers use scripts to create Facebook accounts and join old-style open groups and become fans of pages in order to post spam on the group or page wall. At least with a post on a Page, most people will not see it unless they explicitly visit the page, and hopefully a page admin will remove the spam and possibly ban or report the spammer by then. Also a page admin can set posts from others to be hidden by default, if they don't want to clean up spam. And pages and old-style groups do not have a group chat feature to worry about protecting from spammers.