Race was a convenient taxonomy to classify the different people that Europeans saw when they traveled around the world in the 15 and 16th centuries. Carl Linnaeus, the 18th century Swedish botanist, even assigned various traits: Native Americans (wild), Europeans (gentle, law abiding), Asians (melancholic), Africans (ruled by impulse). Well, you get the drift.
In the 20th century, as I wrote in Outdated Syllabus, anthropologists abandoned race as a valid biological construct; it is no longer used to explain the differences between various peoples. But we all are not the same, you may say. Just walk around the agora, and you will be able to distinguish a Malayali from a Punjabi or a Kashmiri from a Naga. Since there is variation among human population and since we can group people by visible biological traits, doesn’t race exist?
No, says Prof. Tara D Carter categorically in Making of the Modern World 1 course (podcast). When we say people are different, we are referring to their — to throw some jargon —phenotypic trait. It just means a quantifiable trait like skin color or hair color or height. These differences occur due to evolution and these traits are preserved since it helps individuals with the traits survive. For example, dark skin is advantageous in warm climates to withstand the ultra-violet radiation. If you are living in a dense forest, it helps to be shorter. Thus this variation is good for us as a species for survival.
These adaptations do not occur randomly, but is dictated by geography. Over a long period, the advantageous traits become common in a population; the differences among us is just an adaptation. To conclude, biologically we all are the same.