ANSWER: The word "races," other than in the sense of a track and field event, does not appear in most Bible translations. It does appear, however, in the New International Version translation of a verse in Romans 9.
3 For I could wish that I myself were cursed and cut off from Christ for the sake of my brothers, those of my own race (Romans 9:3).
The word translated 'race' in the NIV comes from the Greek word suggenes (Strong's Concordance Number #G4773) and is more accurately translated as 'kinsman' as it is found in the below translation and in others like it.
3. Causing me even to wish myself to be accursed from Christ for my brethren, my kinsmen . . . (Romans 9:3, HBFV)
Scripture does not indicate that Shem, Ham, and Japheth were "white, black and yellow" as many commonly assert. We know that the children of Israel descended from Shem because God's word gives us that genealogy, but there is nothing to indicate that Ham was "black" or Japheth was "yellow" or Oriental. These are suppositions based upon the fact that some of Ham's descendants settled in Northern Africa and one of Japheth's descendants was named, "Javan", which some think might be the antecedent of Japan. However, this is ONLY a supposition or theory and is not something that can be proved from the word of God.
Those whose formal field of study is "races" have proven that many of the North American "Indians" are descended from Asiatic people. The Scriptures are not, however, a historical record of the world and its people. After the first ten chapters of Genesis, the rest of scripture is primarily concerned with the descendants of Abraham and only mentions other people as they encounter God's chosen people.
Much of today's speculation regarding this topic comes in two forms. The first is from those who are racist in their ideology and believe that one group of people is superior to another because of some characteristic like skin color. The second comes from those who want to merge all people together, regardless of cultural differences, out of "political correctness."
How, then, did all the diversity in the human race come from just two people - Adam and Eve? The answer lies in the extensive amount of information that is contained in the human genome. God has placed, in each person's genetic code, a vast amount of genetic variety. For example, even though a person may have black hair and brown eyes he or she may be carrying the genetic information for many other combinations, such as blond hair and green eyes. This is the reason that a husband and wife can have so many varieties of children, such as having short children when they themselves are tall. When we are born our genetic information determines our physical attributes. However, there are many gene-based traits that are not expressed until many generations later, which is why parents sometimes comment about how their children look like their great grandparents.
In the end, according to the Bible, the only "race" that matters is that which comprises all humans. The truth is that ALL mankind has an opportunity to become sons of God through the grace of God and the blood of our Savior, Jesus.