This verse is often pointed to by sceptics as one of those "gotcha" verses, one of the main, well-known "contradictions" which the Bible-thumpers will "never" be able to resolve, or so the sceptics usually treat it. Why this is the case, I'm not really certain, as the application of just a little bit of common sense demonstrates that these verses do not contradict at all, but instead serve to illustrate the fate of Judas Iscariot in a way so as to give a fuller understanding.
To begin, we note Matthew 27:5, which says, "And he cast down the pieces of silver in the temple, and departed, and went and hanged himself." Thus, we see clearly that Judas hanged himself. What then of the passage in Acts where we see Judas falling headlong and bursting like a blood sausage?
"Now this man purchased a field with the reward of iniquity; and falling headlong, he burst asunder in the midst, and all his bowels gushed out." (Acts 1:18)
What is peculiar here is that, when he fell headlong, Judas is said to have burst asunder, basically, his abdomen split open and his internal organs poured out. A rather grotesque picture, and also one which would not happen to a person who was alive only just seconds before. Consider this: A living person must fall from a fairly great height for them to "go splat" as described in Acts 1:18. A fall from the height required to hang a person, such as in a tree or a lamppost, is not sufficient. I had fallen from this sort of a height myself, out of a tree (as have probably many people as they were growing up), and it didn't splat me in this manner, obviously. It hurts, and may even break a few bones, but doesn't do what is described in the verse in Acts. The reason for this is because I am a living, undecayed creature.
The same cannot be said for Judas at the time when he took his tumble. While it is basically impossible for us to presume that Judas fell from such a great height as to splat him while he was still alive (remember, all indications point to his passing away in a field, probably the same potter's field as was later purchased with his blood money), it is certainly likely that his abdominal eruption occurred after he had died and been ripening for at least two days.
Clearly, Judas hanged himself. What happened to his body after this? What NORMALLY would happen to the body of a person who had hanged himself in isolation, in a society which considered death by hanging from a tree to be a curse and greatly ignoble (see Deuteronomy 21:23)? Judas hanged himself on Friday. His body hung from the tree and began to decay. His body would have hung on the tree all day Saturday as well, as this was the Sabbath, and no self-respecting Jew would remove the cursed body of a dead man from a tree on the Sabbath. Thus, his body was hanging for at least two days, if not more, in the hot and humid climate of Palestine in the late spring *. When a body is dead and begins to decay, internal pressure builds up from gas which is generated by anaerobic decay, which begins to swell the abdomen (as this is the softest part of the body, being that it is unprotected and not encased by the skeletal structure). If the rope or the branch broke from which Judas was hanging, his body would fall headlong, and would certainly split open like a ripe melon, right at his abdomen, causing what was left of his internals to spew out, just as is described happening to him. This would be because of the pressure built up from the internal gas caused by decay. If he had been hanging long enough, he may even have popped like a balloon.
Thus, there is a perfectly logical, and in fact probable, explanation for why Judas could both hang himself, and also be described as having fallen headlong and burst open in his middle. The supposition that these two passages contradict each other is based more on the desire of the sceptic for it to be so, than on any actual contradiction which is presented in the written Word of God. Remember, a contradiction is when two written or spoken testimonies say things which cancel out the validity of one another, which is most certainly not the case with these two passages. Instead, in Matthew 27:5 and Acts 1:18, we see two different parts of the same overall story laid out in a complementary fashion concerning the ignoble demise of the traitor Judas.
* - While many Westerners hold to the misconception that Palestine is a terminally arid land, this isn't really the case. Palestine actually has a wet season for half the year, which overlaps with the hot part of the year for that region. This overlap generally is around a month in duration, and occurs in the March-to-May period, right around the time of the Passover and the Crucifixion proceedings. See here for details. Thus, prevailing conditions at the time of Judas' death were likely very wet while also very warm....perfect conditions to promote and accelerate the decay of a dead body.