Who was the Messiah?:
Messiah is a title, not a person. The Hebrew form is mashiah and the Greek form is Christos, commonly rendered in English as Christ. For Jews, the Messiah is the anointed or consecrated one. Kings, priests, and prophets were all anointed in some fashion, but the Messiah is typically regarded as the perfect union of all three: a king, priest, and prophet picked by God to lead his chosen people.
When did the Messiah live?:
Whether the Messiah has ever lived or not is a matter of religious dispute. Pious Jews do not believe that the Messiah has arrived and are awaiting a messianic event for some time in the future. Over the millennia various self-proclaimed messiahs have appeared and attracted Jewish followings. The most famous alleged messiah would be Jesus, also called Jesus (the) Christ.
Where did the Messiah live?:
Messiahs have lived wherever Jews have lived: Palestine, Galilee, Europe, and even America. Messiahs are highly context-dependent: they are only recognized by a community as a messiah if they fulfill certain social, political, and religious expectations. Its not simply a matter of fulfilling prophecy, but also of meeting certain needs that have developed in the Jewish community for example, promising to free them of political subjugation.
What does a Messiah do?:
The expectations of what a messiah does are based in part on what the needs of a community are at the time. Every messiah is expected to take charge of Gods chosen people and lead them into a better, brighter future. How that is accomplished depends upon what is happening to Gods chosen people. The messiah may be expected to drive out the Romans, to lead in the establishment of a new nation, etc.
Why are Messiahs important?:
In Judaism, the messiah is a focus of both political and religious aspirations. The messiah is both a religious and a political leader who is carrying out the will of God here on Earth. If someone is recognized as a messiah, they can command absolute loyalty and obedience from others. Their commands are not merely the commands of another person, but also the commands of God.