What are tag badges?
Tag badges are, well, badges which are awarded to users for achieving a certain score on a specific tag, indicating their knowledge and understanding of that tag.
What tags are eligible for badges?
A tag must appear on a minimum of 100 questions to be eligible for tag badges.
How do I earn each type of tag badge?
Bronze: Have a total tag score of 100 across a minimum of 20 answers
Silver: Have a total tag score of 400 on a minimum of 80 answers
Gold: Have a total tag score of 1,000 on a minimum of 200 answers
What is this tag score you speak of?
A tag score is basically the combined total of all the upvotes (+1) and downvotes (-1) you've accumulated on answers under that specific tag (votes on questions do not count). Acceptance of an answer doesn't have any weight in the score. As well, posts that are in community wiki status or that have been deleted do not count towards your tag score. Tag scores are only recalculated once daily, at 03:00 UTC.
My tag badge can't be revoked, like normal badges, right?
Wrong. Tag badges, unlike normal badges, are automatically revoked upon a tag score recalculation if the tag is no longer eligible to have badges, or you no longer have the required tag score or minimum valid answer count for the badge.
Do tag badges do anything besides allowing me to show off?
The gold version of each badge gives you certain close rights for questions marked with that tag:
You can instantly close as a duplicate any question that was originally asked with a tag you have a gold badge for.
You can instantly reopen any question closed as a duplicate that was originally asked with a tag you have a gold badge for.
You can only close or reopen a given question once (this hasn't changed).