Christian Answer
The Jews were jealous as Jesus was getting more and more popular each day, and what they did not like was that Jesus claimed openly, that he was the son of God. this was unpardonable to the Jews , and worthy of a death penalty. but the Jews had no power to crucify anybody. It had to be by the Romans, so Pilate gave the order to do so.
Jewish Answer
There are two questions here. The first is why Jews had difficulty accepting Jesus' message and the second is why the Jews crucified Jesus. Each will be responded to in succession.
Why did Jews Have Difficulty Accepting Jesus' Message
The assumption that the Christian ascription of Jesus' redemptive qualities (i.e. He is the Christ, the truth, and the way) accords with Jewish principles or understandings of the Messiah is false. It is on account of this major issue that Jews opposed what Jesus was saying and doing, even assuming that the Gospels are accurate or historical materials.
The Jewish Messiah is to be an Earthly King, not an incarnation or union with God. As a result, a person claiming to be God himself is instantly recognized as not being the Messiah. John 5:16-18 asserts that Jews clearly believed that Jesus was articulating that he was the physical progeny of God. Additionally, John 3:16 is at fundamental odds with the Torah's prohibition on Human Sacrifice, its prohibition of blood consumption, its prohibition on cannibalism, and its prohibition on expiation for another's sins. There is no verse in the Old Testament that explains that one of the purposes of the Messiah is to die for sin at all (never mind for the sins of other people).
Jesus failed to perform requirements to be the Messiah and additionally performed actions contrary to the established Commandments. The Bible makes clear that one of the defining marks between a True Prophet and a False Prophet is that no True Prophet will ever deviate from the Law as it was established (Deuteronomy 13:1). A Messiah must also comport to this standard, since he is to be a Holy King. Lists of several of these actions continue below.
Functions of the Jewish Messiah that Jesus failed to fulfill (this list is not exhaustive):
Why did Jews Crucify Jesus
Jews hold that they did not crucify Jesus and are not responsible for his demise. The below examination of the New Testament account and the Jewish discussion will explain what Jews hold actually happened, assuming that Jesus was a true historical figure.
Examining the New Testament Account
The New Testament certainly holds that perspective and paints Pontius Pilate as a man who seems fair and honest. Meanwhile, the rabid Jewish crowd demands Jesus' blood in a persistent and foul way.
Conversely, it seems far more likely in actuality that the reverse occurred. The Romans were probably intent on crucifying him and the Jews most likely begged for long-term imprisonment instead of execution. Of course, this would not be out of any love for what Jesus was preaching and many probably would have agreed that as heretic he should be silenced. However, there are only four acceptable implementations of capital punishment in Judaism that are strongly regulated: decapitation, quick strangulation, quick strangulation with internal burning, and stoning. There is complete rejection of any form of capital punishment that uses piercings to kill or leaves the criminal to suffer for hours on end. Crucifixion is in both categories and would be opposed on those grounds; it remains a uniquely Roman punishment.
It is interesting that Pontius Pilate is painted so dovishly in this part of the Bible as concerns the debate on Jesus' death, but nearly every other Roman source and some other parts of the Gospels paint him as a vile and cruel man. Such a man would have delighted in crucifying a man he perceived as a threat and Jesus was a serious threat to the Roman Empire. The Messiah of that period was understood the way that Jews still understand this concept: an Earthly King who would establish a Jewish Kingdom. Necessarily, therefore, Jesus was going to have to raise an army or commit some political intrigue to achieve this goal. The Romans were likely afraid that Jesus would try to lead a rebellion to free Judea from Roman occupation. This fear would prove justified when another Messianic candidate, Bar Kochba, actually led a temporarily-successful revolt against the Romans that eventually ended disastrously for the Jews. In order to prevent Jesus from taking that power, the Romans pre-emptively sought him out and questioned if he was the King of Jews, i.e. someone actively trying to create a Jewish Kingdom in defiance of Rome. When Pontius Pilate received answers that troubled him, why would he spare the rod here when there was such an obvious threat?
The Catholic Church does not even hold to the line in Matthew 27:25 where Jews take upon themselves the bloodguilt for the Crucifixion. The recent Popes, and the Catholic Church in general, have made clear that this part of Matthew (if not the whole book) is a polemic and that the truth is that Romans executed Christ. Many Protestant Churches (like the Lutheran and Episcopalian Churches) got there even sooner than the Catholics. Since Christian leadership does not hold those lines in the Book of Matthew to be necessarily authentic or worth following, Christian leadership does not even advocate for them.
Examining the Jewish Perspective
Jews do not recognize the Christian claim that they killed Jesus. Jesus was executed by a bloodthirsty Roman Tyrant: Pontius Pilate. However, Paul and Peter could not sell Christianity to the Gentiles while claiming that the Roman people were responsible for the Death of the Savior, so Matthew 27:25 ascribing bloodguilt to the Jews was added to avoid condemning the Romans for their act. Jews, while not major fans of Jesus are not responsible for his death and therefore are not proud or remorseful for it.
Jews disagreed with Jesus quite fervently. The most potent infringement of Jewish Law that Jesus put forward was the consumption of human flesh and human blood as the seal of a divine covenant made through human sacrifice (the eucharist). This violates (according to Jews) the sanctity of humanity (Genesis 9:3), the horror of cannibalism (Leviticus 26:29; Deuteronomy 28:53-57), the prohibition of blood consumption (Deuteronomy 12:23), the prohibition of human sacrifice (Genesis 22:11-12), and the prohibition of transferable expiation - that one person can relieve the sins of another. This is just one of numerous Jewish arguments against Jesus.
There even seems to be a discussion in the Talmud on the issue of Jesus (although it is unclear if this is a later edit or part of the original text). The Rabbis come to conclusion that he is heretic and that he merits stoning in this passage of the Talmud.
However, the act of plotting to actually commit murder is far removed from these discussions and more serious. There are many debates in the Talmud concerning both historical individuals and alleged stories like the Oven of Achnai. This does not mean that those events actually happened, only that the thought experiment of "what would happen under these circumstances" was done. The New Testament makes a number of simple errors about Jewish jurisprudence on such matters. For example, it claims that the Sanhedrin convened on Passover to condemn Jesus. However, the leaders of the Sanhedrin followed the letter of the law to a fault (Jesus even reprimands them for it) and one of the laws is that the Sanhedrin can never meet on a holiday, especially one as central as Passover. Secondly, they allude to a connection between the High Priests and the Sanhedrin in agreeing on what actions to take on Jesus. There was an intense political fight between the High Priests and the Rabbis of the Sanhedrin as to the future direction of Judaism. The High Priests were more corrupt and elitist. The Rabbis were more earnest and populist. There is no reason that the High Priests would not wish to keep Jesus preaching if his populist approach would weaken the appeal of the Rabbis.
Jesus was crucified on a Friday.
No, Jesus was not crucified on an olive tree. He was crucified on a wooden cross.
No, Joseph, the earthly father of Jesus, was not alive when Jesus was crucified.
The Roman empire was in power when Jesus was crucified.
Pontus Pilate sentenced Jesus to be crucified.
There were two criminals crucified with Jesus.
The two individuals who were crucified with Jesus on the cross were criminals.
No. Jesus was crucified to clean the souls of all mankind if you accept that he died for you.
Jesus Christ was crucified in the year 30 AD.
According to the Bible, Jesus was crucified around 30 AD.
No. Jesus was crucified under Pontius Pilate.
Jesus' followers did not want him to be crucified. Jesus, himself, did not entirely want to be crucified, but he saw the necessity of it and allowed himself to endure the pain for the greater good.