The idea of mashiach (messiah) is an ancient one in Judaism

The Jewish idea of mashiach is a great human leader like King David, not a savior

There is much speculation about when the mashiach will come

The Bible identifies several tasks that the mashiach will accomplish

Jews do not believe in Jesus because he did not accomplish these tasks

It is written that:

Before the time of the mashiach, there shall be war and suffering (Ezekiel 38:16)

And:

The mashiach will bring about the political and spiritual redemption of the Jewish people by bringing us back to Israel and restoring Jerusalem (Isaiah 11:11-12; Jeremiah 23:8; 30:3; Hosea 3:4-5). He will establish a government in Israel that will be the center of all world government, both for Jews and gentiles (Isaiah 2:2-4; 11:10; 42:1). He will rebuild the Temple and re-establish its worship (Jeremiah 33:18). He will restore the religious court system of Israel and establish Jewish law as the law of the land (Jeremiah 33:15)

Jews do not believe that Jesus was the mashiach. Assuming that he existed, and assuming that the Christian scriptures are accurate in describing him (both matters that are debatable from the Jewish perspective), he simply did not fulfill the mission of the mashiach as it is described in the biblical passages cited above. Jesus did not do any of the things that the scriptures said the messiah would do.

On the contrary, another Jew born about a century later came far closer to fulfilling the messianic ideal than Jesus did. His name was Shimeon ben Kosiba, known as Bar Kokhba (son of a star), and he was a charismatic, brilliant, but brutal warlord. Rabbi Akiba, one of the greatest scholars in Jewish history, believed that Bar Kokhba was the mashiach. Bar Kokhba fought a war against the Roman Empire, catching the Tenth Legion by surprise and retaking Jerusalem. He resumed sacrifices at the site of the Temple and made plans to rebuild the Temple. He established a provisional government and began to issue coins in its name. This is what the Jewish people were looking for in a mashiach; Jesus clearly does not fit into this mold. Ultimately, however, the Roman Empire crushed his revolt and killed Bar Kokhba. After his death, all acknowledged that he was not the mashiach.

Throughout Jewish history, there have been many people who have claimed to be the mashiach, or whose followers have claimed that they were the mashiach: Shimeon Bar Kokhba, Shabbatai Tzvi, Jesus, and many others too numerous to name. Leo Rosten reports some very entertaining accounts under the entry for meshiekh in The New Joys of Yiddish. But all of these people died without fulfilling the mission of the mashiach; therefore, none of them were the mashiach. The mashiach and the Olam Ha-Ba lie in the future, not in the past.

Source: http://www.jewfaq.org/m/mashiach.htm

According to Rabbi Michael J. Cook, Professor of Intertestamental and Early Christian Literature at the Hebrew Union College, there are ten themes in the New Testament that are the greatest sources of anxiety for Jews:

1. The Jews are culpable for crucifying Jesus - as such they are guilty of deicide

1. The tribulations of the Jewish people throughout history constitute God's punishment of them for killing Jesus

1. Jesus originally came to preach only to the Jews, but when they rejected him, he abandoned them for gentiles instead

1. The Children of Israel were God's original chosen people by virtue of an ancient covenant, but by rejecting Jesus they forfeited their chosenness - and now, by virtue of a New Covenant (or "testament"), Christians have replaced the Jews as God's chosen people, the Church having become the "People of God."

1. The Jewish Bible (the so-called "Old Testament") repeatedly portrays the opaqueness and stubbornness of the Jewish people and their disloyalty to God.

1. The Jewish Bible contains many predictions of the coming of Jesus as the Messiah (or "Christ"), yet the Jews are blind to the meaning of their own Bible.

1. By the time of Jesus' ministry, Judaism had ceased to be a living faith.

1. Judaism's essence is a restrictive and burdensome legalism.

1. Christianity emphasizes excessive love, while Judaism maintains a balance of justice, God of wrath and love of peace.

1. Judaism's oppressiveness reflects the disposition of Jesus' opponents called "Pharisees" (predecessors of the "rabbis"), who in their teachings and behavior were hypocrites (see Woes of the Pharisees).

Rabbi Cook believes that both contemporary Jews and contemporary Christians need to reexamine the history of early Christianity, and the transformation of Christianity from a Jewish sect consisting of followers of a Jewish Jesus into a different religion.

Source: https://en.wikipedia.org/wiki/Antisemitism_and_the_New_Testament

What is striking about Judaism is that argument and the hearing of contrary views is of the essence of the religious life. Moses argues with God. That is one of the most striking things about him. He argues with Him on their first encounter at the burning bush. Four times he resists God’s call to lead the Israelites to freedom, until God finally gets angry with him (Ex. 3:1–4:7). More significantly, at the end of the parsha he says to God:

“Lord, why have you brought trouble on this people? Why did You send me? Since I came to Pharaoh to speak in Your name, he has brought trouble on this people, and You have not rescued Your people at all.” (Ex. 5:22-23).

This is extraordinary language for a human being to use to God. But Moses was not the first to do so. The first was Abraham, who said, on hearing of God’s plan to destroy the cities of the plain, “Shall the Judge of all the earth not do justice?” (Gen. 18:25).

Similarly, Jeremiah, posing the age-old question of why bad things happen to good people and good things to bad people, asked: “Why does the way of the wicked prosper? Why do all the faithless live at ease?” (Jer. 12:1). In the same vein, Habakkuk challenged God: “Why do You tolerate the treacherous? Why are You silent while the wicked swallow up those more righteous than themselves?” (Hab. 1:13). Job who challenges God’s justice is vindicated in the book that bears his name, while his friends who defended Divine justice are said not to have spoken correctly (Job 42:7-8). Heaven, in short, is not a safe space in the current meaning of the phrase. To the contrary: God loves those who argue with Him – so it seems from Tanakh.

Equally striking is the fact that the sages continued the tradition and gave it a name: argument for the sake of heaven, defined as debate for the sake of truth as opposed to victory.

The result is that Judaism is, perhaps uniquely, a civilisation all of whose canonical texts are anthologies of arguments. Midrash operates on the principle that there are “seventy faces” to Torah and thus that every verse is open to multiple interpretations. The Mishnah is full of paragraphs of the form, “Rabbi X says this while Rabbi Y says that.” The Talmud says in the name of God himself, about the conflicting views of the schools of Hillel and Shammai, that “These and those are the words of the living God.”