The messiah will be wiser than King Solomon. He will be a prophet whose prophecy will be on a level approaching that of Moses. He will build the Temple, gather in the Jewish people to the Land of Israel, and re-establish Torah law as the law of the land.

The messiah will teach the way of God:

That king who will arise from the descendants of David will be even wiser than Solomon, and a great prophet, close to [the level of] Moses, our teacher. Accordingly, he will teach the people and instruct them concerning the path of God, and all the gentile nations will come to hear him.

His task is to restore the Jewish royal dynasty, rebuild the Temple, and institute the observance of all Torah commandments:

In the future, the anointed king will arise and restore the dynasty of David to its original sovereignty. He will rebuild the Temple and gather in the dispersed of Israel. In his days, all Torah laws will be restored to their previous state. Offerings will be brought, and sabbatical years and jubilee years will be observed in accordance with all the details set out in the Torah.

We will recognize the messiah, not through miracles that he will perform, but because of the role he will fulfill:

You should not think that the messiah will need to perform signs and wonders, create new physical phenomena in the world, revive the dead, or anything like that…. If a king arises from the house of David who studies Torah and performs mitzvot like his ancestor David did, in accordance with both the Written and the Oral Torah, and he compels all of Israel to walk in [the way of Torah] and to strengthen the breaches [in their Torah observance], and he fights the wars of God, he may be presumed to be the messiah. If he does all of this successfully and overcomes all the nations around him and builds the Temple on its site and gathers the dispersed of Israel, then he is certainly the messiah. If he does not succeed in all this, or if he is killed, it is clear that he was not the one promised by the Torah. Rather, he is like all the proper and worthy kings of the dynasty of David who have died. (Rambam, Mishneh Torah, Hilkhot Teshuva 9:2; Hilkhot Melakhim 11:1–4)

Further reading: For more on the Third Temple, see p. 163.