One of the most common questions we receive at Aish.com is: "Why don't Jews believe in Jesus?" Let's understand why – not to disparage other religions, but rather to clarify the Jewish position.
Jews do not accept Jesus as the messiah because:
  1. Jesus did not fulfill the messianic prophecies.
  2. Jesus did not embody the personal qualifications of the Messiah.
  3. Biblical verses "referring" to Jesus are mistranslations.
  4. Jewish belief is based on national revelation.
But first, some background: What exactly is the Messiah?
The word "Messiah" is an English rendering of the Hebrew word Mashiach, which means "anointed." It usually refers to a person initiated into God's service by being anointed with oil. (Exodus 29:7, 1-Kings 1:39, 2-Kings 9:3)

(1) Jesus Did Not Fulfill the Messianic Prophecies

What is the Messiah supposed to accomplish? One of the central themes of biblical prophecy is the promise of a future age of perfection characterized by universal peace and recognition of God. (Isaiah 2:1-4, 32:15-18, 60:15-18; Zephaniah 3:9; Hosea 2:20-22; Amos 9:13-15; Micah 4:1-4; Zechariah 8:23, 14:9; Jeremiah 31:33-34)
Specifically, the Bible says he will:
  1. Build the Third Temple (Ezekiel 37:26-28).
  2. Gather all Jews back to the Land of Israel (Isaiah 43:5-6).
  3. Usher in an era of world peace, and end all hatred, oppression, suffering and disease. As it says: "Nation shall not lift up sword against nation, neither shall man learn war anymore." (Isaiah 2:4)
  4. Spread universal knowledge of the God of Israel, which will unite humanity as one. As it says: "God will be King over all the world – on that day, God will be One and His Name will be One" (Zechariah 14:9).
If an individual fails to fulfill even one of these conditions, then he cannot be the Messiah.
Because no one has ever fulfilled the Bible's description of this future King, Jews still await the coming of the Messiah. All past Messianic claimants, including Jesus of Nazareth, Bar Cochba and Shabbtai Tzvi have been rejected.
Christians counter that Jesus will fulfill these in the Second Coming. Jewish sources show that the Messiah will fulfill the prophecies outright; in the Bible no concept of a second coming exists.

(2) Jesus Did Not Embody the Personal Qualifications of Messiah

A. Messiah as Prophet

The Messiah will become the greatest prophet in history, second only to Moses. (Targum – Isaiah 11:2; Maimonides – Teshuva 9:2)
Prophecy can only exist in Israel when the land is inhabited by a majority of world Jewry, a situation which has not existed since 300 BCE. During the time of Ezra, when the majority of Jews remained in Babylon, prophecy ended upon the death of the last prophets – Haggai, Zechariah and Malachi.
Jesus appeared on the scene approximately 350 years after prophecy had ended, and thus could not be a prophet.

B. Descendant of David

Many prophetic passages speak of a descendant of King David who will rule Israel during the age of perfection. (Isaiah 11:1-9; Jeremiah 23:5-6, 30:7-10, 33:14-16; Ezekiel 34:11-31, 37:21-28; Hosea 3:4-5)
The Messiah must be descended on his father's side from King David (see Genesis 49:10, Isaiah 11:1, Jeremiah 23:5, 33:17; Ezekiel 34:23-24). According to the Christian claim that Jesus was the product of a virgin birth, he had no father – and thus could not have possibly fulfilled the messianic requirement of being descended on his father's side from King David. (1)
According to Jewish sources, the Messiah will be born of human parents and possess normal physical attributes like other people. He will not be a demi-god, (2) nor will he possess supernatural qualities.

C. Torah Observance

The Messiah will lead the Jewish people to full Torah observance. The Torah states that all mitzvot remain binding forever, and anyone coming to change the Torah is immediately identified as a false prophet. (Deut. 13:1-4)
Throughout the Christian "New Testament," Jesus contradicts the Torah and states that its commandments are no longer applicable. For example, John 9:14 records that Jesus made a paste in violation of Shabbat, which caused the Pharisees to say (verse 16), "He does not observe Shabbat!"

(3) Mistranslated Verses "Referring" to Jesus

Biblical verses can only be understood by studying the original Hebrew text – which reveals many discrepancies in the Christian translation.

A. Virgin Birth

The Christian idea of a virgin birth is derived from the verse in Isaiah 7:14 describing an "alma" as giving birth. The word "alma" has always meant a young woman, but Christian theologians came centuries later and translated it as "virgin." This accords Jesus' birth with the first century pagan idea of mortals being impregnated by gods.

B. Suffering Servant

Christianity claims that Isaiah chapter 53refers to Jesus, as the "suffering servant."
In actuality, Isaiah 53 directly follows the theme of chapter 52, describing the exile and redemption of the Jewish people. The prophecies are written in the singular form because the Jews ("Israel") are regarded as one unit. Throughout Jewish scripture, Israel is repeatedly called, in the singular, the "Servant of God" (see Isaiah 43:8). In fact, Isaiah states no less than 11 times in the chapters prior to 53 that the Servant of God is Israel.
When read correctly, Isaiah 53 clearly [and ironically] refers to the Jewish people being "bruised, crushed and as sheep brought to slaughter" at the hands of the nations of the world. These descriptions are used throughout Jewish scripture to graphically describe the suffering of the Jewish people (see Psalm 44).
Isaiah 53 concludes that when the Jewish people are redeemed, the nations will recognize and accept responsibility for the inordinate suffering and death of the Jews.

(4) Jewish Belief is Based Solely on National Revelation

Throughout history, thousands of religions have been started by individuals, attempting to convince people that he or she is God's true prophet. But personal revelation is an extremely weak basis for a religion because one can never know if it is indeed true. Since others did not hear God speak to this person, they have to take his word for it. Even if the individual claiming personal revelation performs miracles, they do not prove  he is a genuine prophet. All the miracles show – assuming they are genuine – is that he has certain powers. It has nothing to do with his claim of prophecy.
Judaism, unique among all of the world's major religions, does not rely on "claims of miracles" as the basis for its religion. In fact, the Bible says that God sometimes grants the power of "miracles" to charlatans, in order to test Jewish loyalty to the Torah (Deut. 13:4).
Of the thousands of religions in human history, only Judaism bases its belief on national revelation – i.e. God speaking to the entire nation. If God is going to start a religion, it makes sense He'll tell everyone, not just one person.
Maimonides states (Foundations of Torah, ch. 8):
The Jews did not believe in Moses, our teacher, because of the miracles he performed. Whenever anyone's belief is based on seeing miracles, he has lingering doubts, because it is possible the miracles were performed through magic or sorcery. All of the miracles performed by Moses in the desert were because they were necessary, and not as proof of his prophecy.
What then was the basis of [Jewish] belief? The Revelation at Mount Sinai, which we saw with our own eyes and heard with our own ears, not dependent on the testimony of others... as it says, "Face to face, God spoke with you..." The Torah also states: "God did not make this covenant with our fathers, but with us – who are all here alive today." (Deut. 5:3)
Judaism is not miracles. It is the personal eyewitness experience of every man, woman and child, standing at Mount Sinai 3,300 years ago.

Waiting for the Messiah

The world is in desperate need of Messianic redemption. To the extent that we are aware of the problems of society, is the extent we will yearn for redemption. As the Talmud says, one of the first questions asked of a Jew on Judgment Day is: "Did you yearn for the arrival of the Messiah?"
How can we hasten the coming of the Messiah? The best way is to love all humanity generously, to keep the mitzvot of the Torah (as best we can), and to encourage others to do so as well.
Despite the gloom, the world does seem headed toward redemption. One apparent sign is that the Jewish people have returned to the Land of Israel and made it bloom again. Additionally, a major movement is afoot of young Jews returning to Torah tradition.
The Messiah can come any day, and it all depends on our actions. God is ready when we are. For as King David says: "Redemption will come today – if you hearken to His voice."
For further study:

FOOTNOTES

(1) In response, it is claimed that Joseph adopted Jesus, and passed on his genealogy via adoption. There are two problems with this claim:
a) There is no biblical basis for the idea of a father passing on his tribal line by adoption. A priest who adopts a son from another tribe cannot make him a priest by adoption.

b) Joseph could never pass on by adoption that which he doesn't have. Because Joseph descended from Jeconiah (Matthew 1:11) he fell under the curse of that king that none of his descendants could ever sit as king upon the throne of David (Jeremiah 22:30; 36:30). (Although Jeconiah repented as discussed in Talmud Sanhedrin 37a and elsewhere, it's not at all clear from the early sources that his repentance was accepted to the degree that the royal line continued through him. See e.g. Bereishit Rabbah 98:7 that the line continued through Zedekiah.)
To answer this difficult problem, apologists claim that Jesus traces himself back to King David through his mother Mary, who allegedly descends from David, as shown in the third chapter of Luke. There are four basic problems with this claim:
a) There is no evidence that Mary descends from David. The third chapter of Luke traces Joseph's genealogy, not Mary's. 

b) Even if Mary can trace herself back to David, that doesn't help Jesus, since tribal affiliation goes only through the father, not mother. cf. Numbers 1:18; Ezra 2:59. 

c) Even if family line could go through the mother, Mary was not from a legitimate messianic family. According to the Bible, the Messiah must be a descendent of David through his son Solomon (2-Samuel 7:14; 1-Chronicles 17:11-14, 22:9-10, 28:4-6). The third chapter of Luke is irrelevant to this discussion because it describes lineage of David's son Nathan, not Solomon. (Luke 3:31) 

d) Luke 3:27 lists Shealtiel and Zerubbabel in his genealogy. These two also appear in Matthew 1:12 as descendants of the cursed Jeconiah. If Mary descends from them, it would also disqualify her from being a messianic progenitor.
(2) Maimonides devotes much of his "Guide for the Perplexed" to the fundamental idea that God is incorporeal, meaning that He assumes no physical form. God is eternal, above time. He is infinite, beyond space. He cannot be born, and cannot die. Saying that God assumes human form makes God small, diminishing both His unity and His divinity. As the Torah says: "God is not mortal" (Numbers 23:19).
with thanks to Rabbi Michael Skobac - Jews for Judaism