For Groups, how does the classification of "Assigned" and "Available" work?

Available shows you the groups that the contact is not a member of. Assigned shows the groups that the contacts has been assigned to. In the edit contact window you may click on an Available group to move it to assigned, thus assigning that contact to that group. You may also do the opposite to move a contact out of a group. The same holds true for selecting the groups to which your email campaign will be sent.