While originally a hope that the Davidic dynasty would be revived after the fall of Babylon, the Messianic concept developed as a sort of response to the Seleucid period, the events of which lead to a new dynasty in the Hasmoneans.
While the Maccabees did restore liberty of worship to Jerusalem, the ideal peace spoken of by Isaiah, Jeremiah, Micah, and Zechariah that would come via David's dynasty (and perhaps run concurrent with the eschatological Olam Haba or the World to Come) did not develop once the Temple was re-dedicated (the first Chanukah).
Enter the Gemara and the Midrash, rabbinical commentary that would introduce the Jewish tradition of an individual known as "the Messiah." While originally seen as a ruler who would redeem Israel, unlike the Christian concept of "redemption from sin," Jewish redemption is the concept of liberating all Jews to return to Israel to live as freemen. Once seen as a single person, eventually the Messiahship was divided into two personages, a king in David's line and a priestly Messiah.
With the arrival of Jesus of Nazareth and Shimon Bar Kokhba on the world scene, the Messiah concept got further divided. It became solidified in Jesus by Christians who claimed that the Messiah concept was central to the Jews, and fluid after the Bar Kokhba revolt ended in Rome banishing all Jews from Israel.
While the Messiah is still not central to Judaism, the concept remains fluid for Jews, with no uniform theology. For some Jews there is still an expectation for an individual figure in the line of David who will being this redemption about.
Since monarchies have been rejected by most of modern society, some Jews see the Messiah as a concept personified. With the return of all Israel to its native soil, all humanity will "acknowledge God," and peace will reign everywhere. Whether this will involve a religious experience and any supernatural occurrences is still a matter of debate. It is also not agreed upon that any of this has a connection with Olam Haba. The Messianic concept is also still not central to all Judaism to the point that not all Jews today embrace it.