Flair is an image summarizing your accomplishments on Seasoned Advice 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.