See this comprehensive and well sourced paper on the topic by one of our site regulars. Cush was the son of Cham, and the grandson of Noah (Exodus 10:6), and according to the Midrash (Genesis Rabbah 36:7), he was cursed by his grandfather to be dark-skinned.
In terms of the location of the land of Cush, he writes:
"Cush" is commonly translated as Ethiopia, and "Cushites" are commonly understood to have been Ethiopians. However, most scholars agree that the Biblical geographic area referred to as Cush/Ethiopia was not only in the location of present-day Ethiopia, rather it spanned Northwest of present-day Ethiopia, including parts of the areas of present-day Ethiopia, Egypt, Somalia, Nubia, Eritrea, and the Sudan. The Midrash says (Exodus Rabbah,§10) that the plague of the frogs settled a border dispute between Egypt and Cush because the frogs did not pass the border, which showed where exactly the boundaries of each country lied. Indeed, it will be evident later that Cush was near Egypt. Nonetheless, Rashi (to Yoma 81b) translates "Land of Hindu" as the land of Kush. From here one sees that the terms "Kush" used does not necessarily always mean Ethiopia. In the beginning of the book of Esther (1:1), Scripture establishes that Ahasuerus was king from Hodu to Kush. The Talmud records (Megillah 11a) that Rav and Shmuel disputed whether Hodu and Kush are close to each other (Shmuel) or far from each other (Rav). The accepted translation of Hodu is India. If one is to assume that Kush refers to Ethiopia, how then could Shmuel say that Kush was close in geographical proximity to India? Rabbi Yaakov Emden (1697-1776) answers (see Hagahos Ya'avetz to Megillah 11a) that there are two places known as Kush. One "Kush" is in Africa and was Ethiopia, while the other "Kush" is in Asia was close to India. The dispute between Rav and Shmuel was which "Kush" the Torah referring to in describing the boundaries of Ahasuerus' rule. Indeed, an Asiatic mountain range known as "Hindu Kush" runs along the Pakistani-Afghan border.
The essay explores many possible interpretations of Moshe's wife being called a "Cushite". Many explain that she was not actually Ethiopian, but was described as a "Cushite" as an expression of her beauty (see Rashi to Numbers 12:1).