Groups are a different than facebook pages.
To create a group you go to facebook.com/groups/create.php you fill in the necessary information about the type of group you want to make. Then decide if you want an open group for anyone on facebook to be a part of or a closed group for particular network of friends. You can even make it a secret (invite only) group. When you add people to the group they are automatically considered a member of the group. If they do not want to be in the group then they need to go to the group and click on edit to remove themselves
Facebook Groups are more of a personal interaction. So when you post, it comes from you personally. Only personal profiles (people) can be in groups – Pages cannot communicate in groups
When you create a facebook business page the content you post on your page appears that it comes from the business page itself, so that content doesn’t have to be linked to you personally.
One major difference between groups and pages is that pages are indexed (recognized) by search engines like google while groups are not.