Let's begin with the common-sense view. Consider a dog. We all can recognize that a dog is a dog. No matter the differences between different kinds of dogs, whether it's black or white, heavy or svelte, fluffy or sleek, long-tailed or short-tailed, we can still tell that they're all dogs, because these features are only accidental (being mere "attachments") to the critters; there is still something about each dog that makes it "doggy," that it has in common with all other dogs. Aristotle and the medieval philosophers who followed in his general line of thinking, like St. Thomas Aquinas, would call this "something that makes a thing what it is" that thing's "substantial form" or "nature" or "essence." We do still see this philosophy preserved in our everyday language, e.g. "Yeah, it may be missing a leg and been spray-painted bright green, but it's still essentially a dog." What makes a dog a dog, or a cat a cat, or a man a man, is its substantial form, its essence. When speaking of this in terms of how we know things, we would say that there is a universal concept of "dog" that can be equally said of all particular dogs; that is, all particular dogs have a participation in the universal concept of "dog."
But some later medieval theologians were dissatisfied with this notion of substantial form or essence, and they had problems with the notion of universal concepts. There were some who said that though we may use universal concepts as a way to talk about things more easily, this universal concept didn't point to anything real--that though we may talk about the concept of "dog," really, truly, in reality, there are only these particular things that share enough common features that we choose to call them all "dogs." There is no such thing as "dogginess," they would say, only things we choose to call "dogs" for the sake of convenience. Individual things are only collections of characteristics (the "accidents" mentioned above), but there's nothing that ties together all these different strands or "stands under" them (substance --> Latin substantia, sub+stantia = "to stand under"); we simply call things with similar characteristics by the same name. This is the philosophy of nominalism (Latin nomen, nominis, "name").
There is nothing that makes two things each "dogs" unless we choose to call them such. Do you see the consequences of holding this philosophical assumption? It would apply equally to everything that exists, including people: if nominalism is true, then there is nothing that makes two things each "human beings" unless we choose to call them such. There is nothing at the core of us all that makes us the same. "Humanity" becomes a useful fiction which can be discarded when it is no longer useful, an arbitrary category that can be filled with different members as it suits us. So an American plantation owner can declare by his fiat that Africans do not fit in the category of human, and he can enslave them. Nazis can pronounce Jews to be less-than, and exterminate them. Abortionists can term unborn children to be mere "products of conception" and kill them. Suddenly, your gender or sex is not a given, but an option, an "identity" you choose; in the nominalist mindset, there is nothing that makes a man a man or a woman a woman.
Or think of the effect of nominalism in this way. Moral laws can only be set in universal terms, e.g. "It is good for humans to do X, and not good for humans to do Y." "Humans" is a universal term; they are all those things which share "humanity," that is, the essence of what it is to be human. But if we deny that this essence, this nature, exists, we deny that there is anything intrinsically common to humans. If humans have no nature in common, if "humans" is a mere label attached to really distinct particular entities, then we cannot say that anything is universally good or bad for them on account of their "humanity;" we would only be able to say what is good or bad for each of the individuals that we label "human." And who else could determine that other than the individuals for themselves? The door is open for each person to make their own morality.
Of course, we could not have a society in which everyone makes their own definitions of everything, especially of right and wrong. So who makes these determinations in a nominalist society? Whoever has power: physical might, or political sway, or financial backing. The one with power defines our terms, and shapes our reality. The one with power, for all intents and purposes, becomes God.
What a terrifying thought. Most terrifying of all that this is the world we find ourselves in today.