Gmail Groups is a nice way to send an email to a group of people you frequently email.
However in a recent Gmail "upgrade" they made it much harder to use
If you want to make Google's Contacts area have "groups" like the way it used to be you can:
From your inbox, change from Mail to Contacts
Click on "more" at the bottom of the left menu
Click on "Leave the contacts preview"
Or watch this video on how to use groups and how to change it to the old way.