Luke 7.1-10 and Matthew 8.5-13. Both passages give an account of the healing of a centurion's friend or servant, yet there are several differences between the two.
Most of these differences are incidental, and don't influence at all the way the text is understood or interpreted. For instance, Luke reports that the servant is "sick and at the point of death," while Matthew says that the servant is "paralyzed" and "suffering terribly." Who's right? They both are. It's possible for someone "at the point of death" to be paralyzed and suffering terribly. Two people giving a report about the same event will often use unique ways of describing this event. In this instance, Matthew and Luke described the man's illness from their unique point of view.
Most of the other differences in the passage are likewise incidental and easily explainable. There is one difference, however, that is more significant, and more worthy of a deeper investigation: in this account, Luke reports that some friends of the centurion, along with his servants, are the ones who interface with Jesus on the centurion's behalf, whereas Matthew's account says that the centurion himself is the one who comes and talks personally with Jesus. This is an instance where both authors can't be right - either the centurion came and talked with Jesus, or he didn't. And if they can't both be right, then one of them must be wrong. But if one of them must be wrong, then what does that say about the testimony of scripture? If Matthew and Luke botched this story, then what else did they get wrong? Thankfully, an investigation into first-century Jewish culture resolves this issue for us and alleviates the notion that the two gospel-writers contradict one another.
In Jewish culture, servants - when sent on errands for their masters - were to be regarded by those with whom they interacted on their masters' behalf as having the same authority as their master. Even to the extent that a person's servants were a physical representation of themselves. They were to be treated and spoken to in the same way that someone would treat and speak to their master.
Thus, when Matthew reports that the centurion himself came and spoke with Jesus, it is likely that in reality it was actually the centurion's servants. However, in Matthew's Jewish brain, those servants bear the identity and authority of their master, so Matthew records that the centurion himself was present, which is technically correct according to Jewish custom. Luke, on the other hand, is not Jewish (Luke was the only Gentile author of the New Testament), and so he is not familiar with the Jewish custom of servants representing the identity of their masters. Therefore, when he reports on this instance, he takes a literal stance: the servants came on behalf of their master.
This understanding resolves the apparent contradiction. Matthew and Luke both report on the same details, and their accounts agree when we take their cultural customs into perspective.
There are no contradictions in the Bible that I know of that cannot be explained by a careful study and understanding of the context, history, and culture of a passage. The more we study the Bible, the more we can clearly see that it has been perfectly written and preserved for our benefit. Perceived contradictions in the text are just that: perceptions not based in an understanding of the text.
You can trust your Bible, and the story of the healing of the centurion's servant just proves it all the more.