Although the word "race" is found four times in the KJV Bible (Psalms 19:5, Ecclesiastes 9:11, 1Corinthians 9:24, Hebrews 12:1), it is used in reference to running or pursuing a particular course. Most translations use this word sparingly as well. One translation, however, the New International Version, uses it for Romans 9:3 when it states, "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."
Biblical Race
The word translated "race" in Romans 9:3 comes from the Greek word suggenes (Strong's Concordance Number #G4773). The word means a relative (by blood) and by extension a fellow countryman. Thayer's Greek Definitions agrees with this assessment.
A more accurate translation of Romans 9:3 is, "Causing me even to wish myself to be accursed from Christ for my brethren, my kinsmen according to the flesh" (HBFV).
Noah's Sons
Although there are many that believe otherwise, Scripture does not indicate that Noah's three sons, Shem, Ham, and Japheth, each represented a particular race (e.g. white, black and yellow). We do know, however, that the children of Israel descended from Shem because God's word gives us that genealogy. What people commonly refer to as races requires research of non-Biblical sources.
Many suppositions regarding race are based upon beliefs such as some of Ham's descendants settled in Northern Africa. Such beliefs, however, are only theories that cannot be proven directly from the Bible.
Those whose formal fields of study are humans and their migration patterns 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 the Eternal's chosen people.
Two Forms of Speculation
Much of today's speculation regarding race comes in two forms. The first is primary 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 hypothesis comes from those who want to merge all people together, regardless of cultural differences, out of "political correctness."
Diversity
All this, however, begs an important question. How did the diversity we see in the human race occur if everyone ultimately came from the same parents (Adam and Eve)? The answer lies in the extensive amount of information that is contained in the human genome. God made our genetic code to express a large variety of possibilities.
For example, even though a person may have black hair and brown eyes, he or she is carrying the genetic information for the expression of other combinations. This is the reason why a couple can have children with a variety of characteristics (height, hair and eye color, etc.).
Many gene-based traits are not expressed until many generations later, which is why parents sometimes comment on how their children look like their great grandparents.
In the end, based on the Bible's definition, 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 Christ.