(disclaimer, I am Jewish)
The Messiah, in Hebrew "Moshiach", literally translates to "savior". In the books of the Prophets in the Tanach (Old Testament), it was prophecized that the Messiah would come and redeem the Jewish people.
The Messiah was supposed to ride into Jerusalem on a white donkey, and also, seven years before the Messiah was born, there was supposed to be a red heifer born. The red heifer was a kind of cow, a red one, used in the purification rites in the Temple.
The times of the Moshiach, known also as "Ha-Olam Ha-Ba" (the World to Come), would be times of peace, love, and harmony. Also, according to the prophecies, all the dead people will rise and come back to life.
The key distinction between Judaism and Christianity -- besides all the other religious issues; and this is also what originally made early Christians break away from the Jews -- is that Christians believe that the Messiah has already come, in the form of Jesus, while Jews still wait for the Messiah.
The Jews claim that Jesus is not the Messiah, and never was. Some of the evidence for this idea comes from the fact that when the Messiah comes, the world is supposed to be at peace. Look around, is there peace?
However, Christians claim that Jesus will come back in a "second coming."
Some Jews, notably some Reform and Reconstructionist Jews, do not believe in the Messiah.