Why do the Jews not believe that Jesus was the Messiah?
Romans 11:25, "I do not want you to be ignorant of this mystery, brothers, so that you may not be conceited: Israel has experienced a hardening in part until the full number of the Gentiles has come in."
Paul says that the Jews have been blinded to the truth of Jesus as the Messiah to provide for the gentiles salvation. If they would have accepted Jesus at His first coming the Millennial Kingdom would have begun right away.
God has achieved this blindness for the most part through Jewish theology.
The short answer as to why the Jews for the most part reject Jesus as Messiah is, "because He died."
Jewish theology typically looks at passages that you and I as Christians interpret as Messianic (i.e. Isaiah 53, Psalm 22 for example) as referring to all of Israel and how they as people have suffered through the years. There are other interpretations of these passages, but this view as first put forth by the famous middle-ages Rabbi known as 'Roshi' is the most generally held view.
The word "Messiah" is an English version of the Hebrew word, "Mashiach."
Mashiach in Hebrew means "anointed" or "anointed one."
Jews believe that there are many "messiahs" in the Bible since every King of Israel and High Priest was anointed with oil and may be referred to as "an anointed one."
Some modern Jewish scholars believe that the Hebrew word, "HaMashiach" or "The Messiah" does not appear anywhere in the Torah (first five books of the Bible) and was a later interjection during the age of Prophets. However, traditional Judaism insists that this concept was always there in the Torah, though not explicitly mentioned, since Torah was written for a common man to understand in a simple manner. An explicit mention of a future spiritual reward would have been beyond comprehension for the people at that time.
Jews believe that one of the main elements to Biblical prophecy is that there is a yet future age of perfection characterized by universal recognition of God and peace. (see, Isaiah 2:1-4; Zephaniah 3:9; Hosea 2:20-22; Amos 9:13-15; Isaiah 32:15-18, 60:15-18; Micah 4:1-4; Zechariah 8:23, 14:9; Jeremiah 31:33-34) We also believe in this age of peace but refer to it as the "Millennial Kingdom" which will begin at the time Jesus' second coming to end the battle of Armageddon.
Further, Jews believe these portions of Scripture speak of a descendant of King David who will rule Israel during this age of peace. (see, 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.)
Jews feel that the Bible never speaks of believing in the Messiah. They feel that his reign will be an obvious reality and self-evident to any person and will not require faith to believe.
In addition, Jewish theology does not allow for a "first" and "second" coming of the messiah.
Basically, since Jesus did not deliver Israel from bondage and usher in the Millennial Kingdom at His first coming, there is no room for them to believe He is The Messiah in current or traditional Jewish Theology.