Most of the answers are incorrect. The Prophet Moses only had one wife: Zipporah the Medianite. The Cushitic Ethiopian Wife mentioned was only betrothed to Moses the King. Studying the Holy Scripture doesn't mean reading it 50 times, it necessitates and requires relative commentary or commentaries. This is why the Holy or Oral Tradition is absolutely necessary, or the complete meaning of word gets lost. That is how Protestantism in its context destroys Christianity and Apostolic tradition.
In short, Moses in exile fled beyond the Egyptian borders of that time. Yes, there were Egyptian borders at that time. He went into the lands of Cush, Upper Nubia or Ethiopia (as it is known today). At this time, he stumbled into the camp of the Ethiopian King and his army, who were situated outside of the capital city, also exiled from war. Moses ended up strategizing a plan for the King and his forces to recapture the kingdom, but during the battle, the Cushitic king died. Because of his expertise, Moses attained high rank and eventually was made king of these Cushites. As king, he was provided the Cushitic princess, who he refused to sleep with during his 40 year reign. Offended by this, she grew offended and informed the Cushites of his neglect and then he was exiled from the Cushitic kingdom.
The most important topic here is the pseudo-rebuke of the prophet by his sister Mariam, who was endowed with a certain level of prophecy at this time, accused Moses of being married to this Cushitic woman, when in reality, he had no relations with her other than honoring her virginity! The lesson here from God is even if a person has a gift of prophecy, they should use their discretion and gift wisely, not only for intellectual purposes, but for the work and faith of the Almighty. This is why she was struck with leprosy, but cured through intercession of the meek; as it says in the next verses: "Who was meek in the world in the sight of the Lord except Moses his faithful servant?"
You can read detailed commentary of certain rabbis and/or the Ethiopian Orthodox Church's commentary. It has nothing to do with racial inferiority as even some rabbis and most sophomoric readers suggest. If this was the case, how is it that according to Ethiopian and Yemeni tradition, the Queen of Saba, the Cushites have King Solomon's first son? If you don't believe this, the Lord Jesus Christ himself mentions 'the queen of the south: Queen of Sheba/Saba."