BrightID is an application used to uniquely identify individuals. It doesn't use a centralised authority but determines a user's uniqueness using a social graph. BrightID users scan QR codes of other BrightID users and once they have connected to enough users the application will verify them as unique. Users can also flag other users as having duplicate accounts.
1Hive currently requires a user be BrightID verified to claim from the Honey Faucet. In future once released, keepers in Celeste will also be required to be BrightID verified.
For details about BrightID and getting verified see the BrightID website: