Dephaser pulley is the oil filled inlet cam pulley. It controls the VVT (variable valve timing) and means the inlet pulley can become out of phase with the other pulleys by a number of degrees, controlled by the ECU. This then increases or decreases valve overlap duration etc, mainly for EGR (exhaust gas recirculation) and emissions purposes afaik.
But as to why it may fail or be noisy, I would imagine something to do with being starved of oil or something, but not sure. Sorry! At least you know what its for though