Answer:
   
   "Messiah" means annointed. Whenever a line of kingship was
   established, such as Saul, David, or after a disagreement about who
   should get the throne, the king was "crowned" by being annointed with
   olive oil. This was also done for the High Preists from Aaron until
   the end of the first Temple period. Cyrus is also called by G-d, "My
   annointed king". So, when we talk about "a messiah", we are merely
   talking about a king of the Davidic line. Not some kind of
   supernatural entity or one-of-a-kind event. It also means that the
   first mention of the messiah in the Torah would be any mention of King
   David or of Judah maintaining rule beyond the death of Gedalyah (the
   last governor of First Commonwealth Judea), which would imply a
   restoration of the rule. This we find toward the end of Genesis,
   49:10.
   
   Do Jews believe the Messiah will live an eternal life? No. The
   messiah, like all people, will live and die. He will then be succeeded
   by his son (if he has one), and the line of Davidic kings will
   continue. He will be resurrected when the rest of the righteous are.
   But that's (in nearly all opinions) a different era than the messianic
   one.
   
   The different movements via the Messiah differently. The traditional
   opinion was best expressed by [5]Moses Maimonides (RaMBaM), who said
   the following about the Messiah:
   
     "If a king will arise from the House of David who is learned in
     Torah and observant of the mitzvot [the Torah's commandments], as
     prescribed by the written law and the oral law, as David his
     ancestor was, and will compel all of Israel to walk in [the way of
     the Torah] and reinforce the breaches [in its observance]; and
     fight the wars of G-d, we may, with assurance, consider him the
     Messiah.
     
     "If he succeeds in the above, builds the Temple in its place, and
     gathers the dispersed of Israel, he is definitely the Messiah. ...
     
     "If he did not succeed to this degree or he was killed, he surely
     is not [the redeemer] promised by the Torah. [Rather,] he should be
     considered as all the other proper and complete kings of the
     Davidic dynasty who died. G-d only caused him to arise in order to
     test the many, as [Daniel 11:35] states; "and some of the wise men
     will stumble, to try them, to refine, and to clarify until the
     appointed time, because the set time is in the future."
     
   The Rambam then continues by explaining why Judaism has rejected the
   claims of other religions, notably Christianity, that "caused the Jews
   to be slain by the sword, their remnants to be scattered and humbled,
   the Torah to be altered, and the majority of the world to err and
   serve a god other than the L-rd." Since, he said, the required
   criteria [as described in the preceding paragraphs] have not been met,
   all messianic claims to date, such as Christianity or the the beliefs
   of the followers of Shabtai Zvi, have been proven false.
   
   The full text is in his [6]Mishneh Torah, Sefer Shoftim, Hilchot
   Melachim U'Milchamoteihem, Chapter 11. This translation was done by
   Rabbi Eliyahu Touger, published by Moznaim Press, from Halacha 4.
   
   The Rambam's statement is probably the definitive rendering of the
   traditional Jewish view on the subject. Others believe that the
   Messiah will usher in an age of miracles, and will come in a
   miraculous manner.
   
   The liberal movements, such as [7]Reform) do not believe in a personal
   messsiah, but do believe in the concept of a messianic age.