If Private is in the name, it means group members must be invited or added by the group captain. The group is not searchable or viewable by anyone outside the group.
If Searchable is in the name, it means the group can be searched for, and people can request to join the group.
If Community is in the name, then only members of the given Community, Club, School, or other Organization can be in the group.
If you're forming a group and your club or community is not yet a GroupValet customer, then your only option will be Private. If your organization is a GroupValet customer, then you'll have the option of choosing what type of group you'd like to create.
One more note - members of Private groups need to be invited to the group and they are not members until they accept the invitation. However, since Community groups are tied to a roster from the parent organization, members of Community groups can be "added" to the group without the need for accepting an invitation. So the process for adding members is much easier for the Community groups.
Private groups require a subscription to GroupValet. Community Private and Community Searchable groups do NOT require a subscription as they are covered by the parent organization's license.