Short answers, pulled from the story.
Messiah is a transliteration of the Hebrew word mashiach, meaning "anointed," from the verb meaning "to rub with oil, to anoint." The Greek translation is Khristos, anglicized as Christ, which carries the identical meaning. In Hebrew, the Messiah is formally called melekh ha-mashiach, meaning "the Anointed King."
In Judaism, the Messiah is a fully human, non-divine Jewish leader physically descended through an unbroken paternal line from King David and King Solomon. He is expected to unify the tribes of Israel, gather all Jews to Eretz Israel, rebuild the Temple in Jerusalem, and usher in a Messianic Age of global universal peace. Belief in his eventual coming is one of Maimonides' 13 Principles of Faith.
Christians believe Jesus of Nazareth fulfilled the Old Testament messianic prophecies in his mission, death, and resurrection, with the remaining prophecies to be fulfilled at a Second Coming. Mainstream Christian theology holds Jesus to be the Son of God and God the Son, a concept fundamentally different from Judaism's view of the Messiah as a fully human political and religious leader who is not divine.
The Quran identifies Jesus (Isa ibn Maryam) as the Messiah and a prophet sent to the Children of Israel, but not as God or the Son of God. Islam holds that Jesus did not die on the cross but was raised bodily to Heaven. According to Islamic tradition, Jesus will return at the end of times alongside al-Mahdi, defeat al-Masih ad-Dajjal (the false Messiah) at the Gate of Lud, and live on earth for forty years before dying.
Mirza Ghulam Ahmad (1835-1908) was the founder of the Ahmadiyya Movement, who declared himself the promised Messiah and Mahdi following divine revelations in 1891. He argued that Jesus had survived the crucifixion and died a natural death in the east, allowing for a spiritual rather than physical second coming fulfilled in himself. His 1880 work Barahin-e-Ahmadiyya drew on Indian, Sufi, Islamic, and Western thought to defend Islam against British colonialism and missionary activity.
Many Chabad Lubavitch followers believe Menachem Mendel Schneerson (1902-1994), the seventh Rebbe of Chabad, is the Messiah. A Chabad halachic ruling declared it incumbent on every Jew to believe Schneerson is King Moshiach. Thousands of visitors and letters arrive each year at his gravesite, the Ohel, on the anniversary of his death.