Bavel – בבל (Babylonia) is what Jews called the region of present-day Iraq. In the Talmudic period it included Persia and was ruled in Sasanians. Later, in the period of the Geonim, by the Abbasid Muslim caliphate.
גאונים – sing. גאון (Gaon) – the formal title of the head of one of the yeshivot (academies) of Bavel (Babylon, or present-day Iraq), which was also known as Rosh Yeshiva (Gaon Yaakov) or Reish Metivta. The office itself is referred to as the gaonate and stood in contrast to the Reish Galuta, “Head of…