The Groups tab found on your own profile page will show you a list of public member groups, as well as any member groups that your member account is currently a member of. On this page you'll see the name of a member group, how many leaders the group has (if any), the amount of members apart of the group, and the status of the group letting you know the type of member group it is.
Depending on how a member group is set up, some member groups can be joined directly from the groups page of your own profile. If available, you can click the Join Group link listed below the member group name on the page. Alternatively, if you're part of a member group and wish to no longer be in the group, visit the Groups tab of your profile and then click the Leave Group link found below the title of the member group that you are part of.
If you're viewing the profile of another member, you can find out which member groups the member is part of by clicking on the Groups tab as well. Since it is possible to be a member of multiple groups, you may see multiple groups listed on the page for the profile of the member that you are viewing.