According to the contemporary definition, a homology is something like a family resemblance. Its a similarity that indicates two or more organisms are related to each otherthat they share a common ancestor.
The authors of Science and Creationism: A View from the National Academy of Sciences explain it like this: [T]he skeletons of humans, mice, and bats are strikingly similar, despite the different ways of life of these animals and the diversity of environments in which they flourish. The correspondence of these animals, bone by bone, can be observed in every part of the body, including the limbs; yet a person writes, a mouse runs, and a bat flies with structures built of bones that are different in detail but similar in general structure and relation to each other.
Scientists, they add, have concluded that such structures are best explained by common descent.
Homologies differ from similarities that are not acquired from a common ancestor. Thus, the eyes of humans and octopi are very similar, but scientists do not think their common ancestor had such an eye. Such similarities are called analogies.
But using the contemporary definition of homology as evidence for common ancestry is circular reasoning. How do you know that two organisms share a common ancestor? Because they have features that are homologous. But how do you know the structures are homologous? Because the two organisms share a common ancestor.
Leaving aside the problem of circularity, it is far from clear that similarities, as such, are best explained by common descent. If we knew there were a mechanism that could produce humans, mice and bats from a common ancestor, that claim would be plausible. But the mechanism is the very thing in question.
In the absence of a mechanism, the fact of similarity does not compel a Darwinian explanation. After all, we see similarities between different kinds of cars, but we dont conclude that one descended from another.
Moreover, biologists knew about homologous similarities well before Darwin published his theory, yet the great majority concluded that they resulted from a common design rather than common descent.