Honestly, the modern concept of race as being something on your blood that is inheritable and you cannot change starts primarily with how Christians in Spain thought about Jews and Moors during the Inquisition -- and because it emerged out of a social and political construct, it was destined to eventually be revealed as such. Purity of blood became a massive obsession, and "Jewish blood" was thought to taint a family in an indelible way -- previously, Jews who converted to Christianity weren't seen as being really inherently different. But the Reconquista and the Inquisition changed all of that, and Jews became a race of people, distinctly, biologically different.

With the emergence of sub-Saharan Africa as a source for slaves, the emergent idea of race came in handy once again -- Africans were an entirely different race of people, inherently different, more suitable for servitude than Europeans. Pseudo-scientific studies were done to classify racial types and to make sense of the schema, but it essentially boiled down to whomever was different from Western Europeans and proving a challenge in their goals of colonialism, imperialism, and profit, they because a different race. For a long time, Southern Italians were a different race. The Irish were a different race. Slavs have gone back and forth between being white and being non-white. The wiggle room is because the concept itself was always malleable so that it could be adapted to suit whatever imperialist need arose. With the 19th century obsession for cataloguing all things and the emergence of eugenics at the end of the 19th century, the idea of race as a scientific dogma was firmly entrenched, leading to a lot of the genocidal violence of the 20th century and complicating the struggles for racial equality and anti-imperialist movements around the world.

Enter DNA. With the isolation of DNA, we no longer had to look at pysiological differences and measure those to say to what degree human beings are like one another. DNA shows it clearly -- and the difference genetically between people of different races is anywhere from 0.01-0.05%. That's infinitesimal. When that scientific reality collided with the actual voices of marginalised people emerging in academia, the idea of race as science began to evaporate. There are still some legitimate questions about race and genetics, race and medical history, what traits are passed down in what social clusters -- the predominance of sickle-cell anemia in Black populations in the US, Tay-Sachs disease in Jewish populations, etc -- but the notion of race as a scientific thing to categorise has dropped off the map because, scientifically, there isn't anything to categorise anymore. The discussion has shifted to difference in culture, history, and experience, and has moved to the social sciences instead, where race is an experience and an ancestry, not some indelible blood marker. We're increasingly realising that race isn't science. It's a way people abused other humans and justified it with science, and the more we're aware of that, the less scientific racism has a hold. It lingers on in other places, but in the West, race as a science has been largely dead since DNA took over as our main means of understanding the differences between humans.

A good overview of all of this, the rise and fall of racism as a concept, is the tidy and readable: Fredrickson, George M., and Albert Camarillo. Racism: A Short History. Princeton: Princeton University Press, 2015.