This is one of the most well-known problems for Biblical inerrancy. Matthew's and Luke's gospels both trace the ancestry of Joseph (husband of Mary, mother of Jesus) back to King David, and Luke continues all the way back to Adam. But their genealogies are completely different.
The standard reply seems to be that one or other of the genealogies is actually of Mary, not Joseph. Yet it's unclear why the text can't just explicitly say this. Countering Bible Contradictions explains that the genealogy "would lose all appeal if it was explicitly cited as Mary's" because people in those days didn't like tracing ancestry via women. But there are a lot of things in the Bible that aren't appealing, and this claim suggests that God is happy to deceive us for the sake of making the text more popular.
Nevertheless, when one considers the idea of a virgin birth, it becomes extremely odd why anyone would bother with a genealogy of Joseph at all, since he's not actually the father of Jesus. For this reason, I don't see it as totally impossible that Luke intended to give a genealogy of Mary. There are some speculative ways of translating the text that make this explicit. For example, Countering Bible Contradictions suggests a translation by Robert Gromacki:
This positioning of the brackets is supposed to indicate that Joseph is not the son of Heli, and thus that we are in fact reading a genealogy of Mary. However, almost no Bible translation that I know of does it this way (but see the CEV for an exception).
A more involved way of getting round the problem is to claim that one of the genealogies goes via Joseph's biological father, while the other goes via an adoptive or step-father. I suppose this is possible, though there is no other evidence for Joseph having a step-father, and I would question whether a competent author would include step-parents in a genealogy.
Another reason for treating Matthew's genealogy as dubious is that it is too short. This is dealt with elsewhere.
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