I suppose that Umayyads, Ummayads or Ommayads are the first dynasty (reigned until 749) of the Islamic empire after the four caliphate era. They are followed by Abbasids (which are different than Abbadids) holding the caliphate between 749-1258. As far as I know Abbadids are not a continuum of Abbasids dynasty and an entirely separate Arab dynasty ErdemTuzun.
When the Abbasid caliphs took over, one of the Ummayads escaped to Spain, and ruled it as an emirate (province). Later it became an independent caliphate. Eventually, though, political power fell to various local rulers, and so the caliphate disintegrated into petty states, of which the Abbadids are one. The same sort of thing had happened in the east, too - by 800 it makes more sense to talk about emirates than caliphates as political units.