Bronze and silver tag badges confer no privileges.
If you have a gold tag badge then you earn some privileges related to duplicate closures.
- The Dupehammer: When a question has your gold-badge tag, your votes to close it as a duplicate are unilateral. This means it is immediately closed as a duplicate of whatever you said because you said so. This is like what diamond moderators can do except only for duplicate votes. Your dupehammer doesn't work if you were the one that added your gold-badge tag, or if you approved the edit that added it, in order to prevent funny business. (See uno, dos.)
- Your reopen votes are also unilateral if the question has your gold-badge tag and is closed as a duplicate. (This is the reopen hammer, or the un-dupe-hammer. Its name is less cool.)
- You can edit the duplicate targets of a question closed as a duplicate. You may add or remove which questions this question is a duplicate of.
You have these powers because once you've earned a gold tag badge you have seen a lot of questions within that tag, and you probably know very well what's a duplicate of what.
Those who earn a gold tag badge find it takes some getting used to: remembering they have it, and figuring out how to use it well.