Most interpreters believe the Cushite woman is Zipporah, the wife of Moses. She was from Midian (Exodus 2:16–22). They see Cush as referring to Cushan, a region and people near Midian (Habakkuk 3:7).
Other interpreters think Cushite refers to Kusi in northern Arabia, Ethiopia, or Sudan. This would suggest she was a different wife.
In either case, it is possible that Miriam and Aaron disapproved of his having married a non-Israelite or that they were jealous of her influence over Moses.