And throwing down the pieces of silver into the temple, he departed, and he went and hanged himself.
(Now this man acquired a field with the reward of his wickedness, and falling headlong he burst open in the middle and all his bowels gushed out.
The Claim: Judas hung himself vs. Judas died by falling and bursting open.
The Explanation: Usually when Christians are presented with this contradiction, they respond by saying "why can't it be both?" That response does not satisfy me, and I'm a Christian, so if it doesn't satisfy me, how could anyone expect it to satisfy non-christians? Therefore, I'm going to take things a step further and explain how it can be both, because if it's possible that it could be both, then it's not a contradiction.
Let's start in Matthew 27:5, where Judas went and hanged himself. Ok, that's fine so far. Now, I'm sure some time passed between Matthew 27:5 and Acts 1:18. This meant that Judas was out hanging from a tree (or whatever he used to hang himself from), and Judas was out in the sun, and the wilderness. Judas's dead body would then begin to decay and rot.
Now we go to Acts 1:18, where Judas's body fell to the ground. His body could have fallen by the rope breaking, being cut, the tree branch breaking or being cut, or whatever (the Bible doesn't get that specific). Now when a rotting and decaying body falls to the ground, it would make sense that the body burst open, with blood and guts and intestines spilling out everywhere.
In general, a healthy, living body would not just burst open with blood and guts spilling out everywhere from a little fall. You can verify this yourself (and probably have) whenever you've fallen down. Or you can watch kids, they fall down all the time. They fall down by tripping while walking/running, they fall down the stairs, they fall down the slide, etc. Unless there's some heavy impact made by falling a long distance or they land on something sharp, the kids generally aren't bursting open with blood, guts, and intestines flying everywhere. Judas's body however, was dead and decaying when it fell down, so having the body burst open makes sense.
It's also possible that Judas hanged himself over a cliff, so that when his body fell in Acts 1:18, his body burst open due to the high impact. It's not required to know all of the intricate details of Judas's death. All that is required to show how both verses are possible to co-exist. If it is possible for both Matthew 27:5 and Acts 1:18 to be true, then there is no contradiction.