For almost every user, the Member Group is not displayed. In your case, you are not in a membergroup and therefor have "regular user" access.
The Post group is based on number of posts in valid categories and the grouping goes in order: Muggle, Newbie, Geocacher, Power Cacher, Crazy Cacher, & Geocaching Addict
The icons below that are also based on number of posts. Muggles get the DNF face and Newbie on up get an increasing number of smileys. Users in Member Groups get whatever I feel like giving them

There is also a "Title" field that displays. Mine is "DNF King" and I added one to your profile, too. Click "Profile" then "Forum Profile Information" and look for "Custom Title". If you don't see any of those options I may have disabled it.
The other icons are to view profile, view website, send e-mail, send private message, etc.