There are quite a few accounts. There is a comprehensive list on wikipedia, and a list with more information but fewer entries in the Jewish Encyclopedia. Some Jewish sects — especially Hasidic ones — believe that there is a potential Messiah in every generation (this is Chabad's view, for example).

Bar Kokhba (- 135 CE), the leader of a large revolt against the Romans, was believed by Rabbi Akiva and many others to be the Messiah.

The most infamous Messiah claimant was Sabbatai Zevi (1626 - 1676), who was forced to convert to Islam by the Ottoman Sultan.

Most recently, the Lubavicher Rebbe Menachem Mendel Schneerson was regarded by many of his followers as the Messiah until his death in 1994. A small sect still continues to believe that he was the Messiah and predicts his return. If you want to learn more about him and his movement, I recommend reading The Rebbe's Army by Sue Fishkoff.