What is a basic model for making a database with users and groups?
The traditional way to model this is using a pattern called Role-Based Security.

The idea is not just to have groups of users, but also groups of permissions. Here is how the pattern looks:



Note that you want to avoid reserved words for table names, so don't name your tables exactly as shown in the diagram.  

The way it works is that your groups or Roles have not only a list of users assigned to them but also a list of permissions assigned to them.  This allows you to table-drive both who can do what, but also what it is they can do, if you follow me.