User groups allow you to control which functions of your store various users can access and/or modify. For example, if you have someone that you only want to be able to modify the design of your store, you could create a user group called "Design" and then only allow them to access the design related pages. If they tried to click on the Payments page, they would be told they didn't have authorization.
Setting up these groups is easy. Just navigate to System -> Users -> User Groups. You will see any groups you've set up, as well as a Top Administrator group (the default, which is access to all features). To add a group, just click the "Add" button in the top right corner. You can now chose a name for this user group, and then check what they can access and what they can modify. Be sure that if you give a group permission to modify a page, they also have permission to access it! Once you have selected all the pages you would like them to access/modify, just click save! You can add as many user groups as you need.
Once your user groups are set up to your liking, you can assign each user to a group by editing their user page (System -> Users -> Users) and selecting the group from the drop down menu and clicking save. It's that easy!