To start out, you can access the Member Groups page by clicking the "Member Groups" link found in the header (shown below):
As a normal XMB user you will either be added to a member group by an XMB or Group admin, or you can apply for membership in any group accepting applications.
As you can see in the image above, the group "Group2" is currently accepting applications. You may apply to join the group, simply by clicking the Apply link. After clicking the link, you will automatically be added to a list of applicants for that group. You may only apply to a group once. Once you've applied, your application must be approved by a grouop administrator. A group admin has the option of Approving or Denying your application. If your application is Denied, then your name will go on a list of rejected applicants, and you will not be able to apply to that group again until your name is removed from that list by a group admin.
When an admin has made a decision on your application, you will recieve a U2U informing you whether you have been approved or denied. If your application has been approved, you will automatically be added to the group's userlist, and have access to any areas that that group has specific access to.
Now as a group member, you have the ability to contact all other group members via E-Mail or U2U.
Returning to the index, you can now see that you have access to the previously hidden "Group2's Forum". You would have access to this forum as long as you remain a user in Group2.
Note: Groups can have the option of a minimum post limit before one can apply to a group. If the group you're interested in doesn't have an apply link, this may be the case. You may want to talk to member of that group and see if they have a post limit in place.