Flair is an image summarizing your accomplishments on Genealogy & Family History Stack Exchange or the Stack Exchange network (your reputation, your badges, the top sites you've participated on), suitable for display on your own website – or printing out and pinning to a vest, if that's what you prefer.
Example:
To see what your flair looks like and how to use it, log in and go to the "Flair" page on the Settings tab of your profile.