This is one of the most well-known problems for Biblical infallibilism. 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.
Countering Bible Contradictions also suggests a speculative translation of Luke 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, no Bible translation that I know of does it this way. Nevertheless, there is another reason to believe that Luke might in fact be recording Mary's ancestry. This is Luke 1:32:
"Father" here means "ancestor". This verse appears to indicate that Jesus was a descendant of David, which is only really true if Mary is a descendant of David (assuming the virgin birth is true, of course). This would give Luke a strong reason to write a genealogy of Mary. I hadn't noticed this when I first wrote this page, and have therefore downgraded the problem from Serious. I'll confess I really wanted this one to be serious.
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.