There are a few incidental mentions of race in the Bible (e.g., that Ethiopian's skin was different, Jeremiah 13:23), but there is nothing saying one race is superior to another. Moses was married to a Cushite (Ethiopian) woman (Numbers 12:1-16), and God was angry with Aaron and Miriam for criticizing that marriage.
The Hebrews were forbidden to marry Canaanites and other pagan peoples (Exodus 34:11-16; Deuteronomy 7:1-3). The reason was because those people worshipped idols, not because of race; the Hebrews, Amorites, Canaanites, etc. were racially similar.
The apostle Paul encouraged Christians not to marry unbelievers (2 Corinthians 6:14), but here again, the reason was religious, not racial.
Jesus and His apostles taught that we must respect and show compassion for all people of God's creation without regard to artificial distinctions like race and nationality. In His Parable of the Good Samaritan (Luke 10:25-37), Jesus told of a Samaritan man who was kind to a Jewish man even thought they were enemies of different religions, nationalities and racial backgrounds. At the end, Jesus tells us, "Go and do likewise."
The apostle Paul stated the equality of all people without regard to racial/ethnic background, social status, etc.:
You are all sons of God through faith in Christ Jesus, for all of you who were baptized into Christ have clothed yourselves with Christ. There is neither Jew nor Greek, slave nor free, male nor female, for you are all one in Christ Jesus. (NIV, Galatians 3:26-28)
A number of attempts were made to use the Bible to justify those bans on interracial marriage. Vague assertions were made that God intended for the races to remain separate. Some verses (Exodus 34:10-16, 2 Corinthians 6:14, etc.) were quoted in part or otherwise out of context in an attempt to show that God opposed interracial marriage.
Related article: What Does the Bible Say About Marriage, Divorce and Remarriage?