A very common refrain I heard growing up is that there were no mistakes in the Bible; there were no contradictions. Anything that people presented as being a contradiction was simply because that person hadn't really thought about it long enough on how it actually isn't a contradiction.
However, despite my quite rigorous education in the Bible, I have become aware of a number of problems with the idea of there being no errors in the Bible. Most of the time, I can only assume that either the people teaching me were unaware of these problems, or decided that these problems were not worth teaching me. Perhaps for some of you reading this, this may be the first time being presented with some of these.
Judas' Death
For example, how did Judas die? In Matthew we have this recorded:
In Acts we have this statement:
There are 2 problems here. The first is that in Matthew Judas dies by hanging himself, but in the Acts Judas falls and his intestines fall out. The second problem is that the 2 stories give completely different reasons for why the field was called the "Field of Blood". Matthew says it is because the counsel bought the field with Judas' "blood money", but in Acts the field is called "Field of Blood" because Judas spilled his guts all over it.
People as early as Augustine of Hippo (354-430) explained the discrepancy of Judas' death as simply Judas hanging himself, the rope snapping, and then his guts splaying open. Basically, he created a story that was able to explain the details of both passages. For some people, this explanation is sufficient to their own satisfaction. Though personally I this feels lacking to me because people don't really "burst open" from a short fall, but also the verse specifically says that he fell "headlong". I don't know how someone's rope snaps from hanging themselves and then the fall head first. Also that Acts explicitly says that he fell, and just happened to forget the detail that Judas was hanging himself? The details on this explanation simply don't add up in my estimation.
However, even if we accept Augustine's story at face value, then we are still left with the problem of the 2 books giving different explanations for the "Field of Blood". I am sure that someone out there has attempted to explain this discrepancy, but I have not been able to find any explanation.
Jesus Quoting Samuel
Here Jesus is referencing a time in scripture where David eats the show bread in order to make a point about his disciples picking and eating grain on the Sabbath. The reference Jesus uses is found in 1 Samuel 21.
The main problem here is that Jesus says that the high priest's name was Abiathar, but we can see in the story that the high priest at the time was Ahimelek, Abiathar's father.
Like other passages, you can google around for articles and long explanations (I have not found any short ones) on how to explain why Jesus uses Abiathar over Ahimelek, and I encourage everyone to do so for their own education. One explanation requires stretching "in the days of Abiathar the high priest" to also include the days before he was actually high priest. Another explanation is that Jesus is actually making a play on words by using Abiathar instead of Ahimelek on purpose, and he is actually intending to emphasize his rejection of the temple priesthood of his time.
Perhaps, it was just a mistake.
When was Jesus Crucified
We have another interesting distinction here between the different books. In Mark, it says that Jesus was crucified after they ate the Passover meal, but in John it says that he died on the day before the Passover.
There are explanations out there that make the case that John and Mark are using different calendars, or that Passover was really 2 days long, or that the meal Jesus and his disciples had in Mark wasn't really a proper Passover meal.
Perhaps, it was just a mistake.
Paul's Destination after Conversion
In Galatians, Paul is telling a brief version of his conversion story. He says that he explicitly did not go to Jerusalem but instead went to Arabia and then returned to Damascus. However in Acts, after Saul's conversion on the road to Damascus, Saul spends a few days in Damascus and then the next place it says he visits is Jerusalem.
Taking these to statements at face value, they don't seem to be reconcilable. Acts states that Paul went somewhere that Paul in his own letter explicitly states that he did not go. I have no doubt that there are explanations that you could find online to help harmonize these 2 statements. But like all the previous ones: perhaps, it was just a mistake.