Jesus Christ was a Jew and Christianity originally emerged out of Judaism. It's no exaggeration to say that, without Judaism, there would be no Christianity. Indeed, Christianity was at first not a separate religion at all, but rather a cult within Judaism. It was only after the completion of Christ's earthly ministry that Christianity began to develop into the religion we know today, reaching out beyond its Judaic origins to embrace the world beyond.
Since then, the relationship between the two great world religions has often been fraught, to say the least. For centuries, Christians believed—and were encouraged by their churches to believe—that the Jews had been responsible for Christ's crucifixion. This distortion of history and the Gospel was used by both secular and ecclesiastical authorities alike to justify persecution against Jews, whether official or unofficial.
As regards to what Jews and Christians believe there are some similarities. Jews and Christians worship the same God, share a belief in a covenantal relationship between God and man, and accept the importance of assembling together for worship. That said, the differences are much greater. For one thing, although Jews tend to regard Jesus as a great Jewish teacher, they don't believe that he is the Messiah, that he is the Son of God. The traditional Jewish understanding of the Messiah is as a great warrior king, sent by God to restore David's kingdom to its former glory. Clearly Christ, with his message of universal peace and brotherhood, wouldn't fit that description.
Also, Jews don't accept that the New Testament was inspired by God or that it fulfills, as Christians believe it does, the prophecies set out in the sacred scriptures of Judaism, or the Old Testament as Christians refer to them. Judaism is a more worldly creed than Christianity, focused as it is upon the world around us and the place of believers within it. Christianity, on the other hand, is other-worldly, pointing the way towards a spiritual world in which the devout may receive the gift of eternal life.