Tag Groups are another layer to organizing content with tags. It's unlikely that you'll use Tag Groups unless your web developer employs them for a custom application. Please ask your developer to document specific use of your Tag Groups, as there is no standard application.
An example use for Tag Groups might be a dinner menu, where normal tags are used to categorize food like "salad", "soup" "meat", or "cake", etc. A Tag Group might then be used to organize tags into courses, where "Appetizers" might be a Tag Group with "salad" and "soup" within that group.
Another example might be a staff directory where people are tagged with their working group like "business development" or "payroll" while Tag Groups may be used for departments like "Sales" or "Human Resources".
Tag Groups are a way to hierarchically organize tags while retaining their wonderful nebulous characteristics for categorization.