Barrows amulets are obtained as a reward after finishing Missing, Presumed Death. Thirty are obtained from the quest. You can use your invitation box to teleport back to the Empyrean Citadel and search the statue there after the quest for the amulets.
The amulet can be used to skip certain Barrows brothers in the Barrows minigame by using the amulet on the sarcophagus. Up to three brothers can be skipped during each run. Barrows amulets cannot be used against the Barrows brothers in Barrows - Rise of the Six, nor can they be used against Linza the Disgraced.
When used on one of the sarcophagi within the Barrows, you receive the message "You hear a cry from within the sarcophagus as you shatter the amulet", and the Barrows brother's name appears under the list of Brothers slain.
The Barrows amulet can also be used for the brother guarding the tunnels. Simply use the amulet on the Treasure Chest and it will kill the brother, yielding the message "A cry echoes around the tomb as you shatter the amulet."
However, in order to use the amulet on a brother, he must not be physically present. If you accidentally spawn him, you must make him disappear (running out of a crypt for non-tunnels and running far away from the brother in the tunnel) before the amulet can be used. Using the amulet on the brothers themselves has no effect.