The Flames of Regency is a corona of mystic flame that adorns the head of whomever is the present ruler of the Dark Dimension. The Flames of Regency are associated with the Flames of the Faltine - the sentient self-perpetuating energy that comprises the entirety of the Faltine dimension. Dormammu, Umar and Clea are all aspects of the Faltine.
In most cases, the ruler of the Dark Dimension is the dread Dormammu, the Flames of Regency enveloping his features entirely. Dormammu's sister Umar inherited the Flames during her reign as regent. Umar's niece Clea, likewise took hold of the Flames of Regency when she usurped the throne from Umar.