Allah, the one true God
It is important to note that the Koran is a religious text, and the word 'Character' may not apply. A better word would be "Person" because this section includes historical persons, and although many people don't even believe in any religion or concept of God, Allah is presented in the Koran to be similar in nature to a person, although Allah would of course be infinitely higher in nature.
Allah is represented by the Koran to be the one and only God, responsible for the creation of all things, and the sole divine influence in human history, which the Koran teaches will culminate in a final apocalyse.
Muhammad the True Prophet
Muhammad is the prophet to whom Allah gave the Koran in its Arabic form through prophecies that Muhammad experienced over the course of more than twenty years.
Muhammad's writings are authoritative in the Muslim faith, but his supremacy is different in nature than the Christian belief in Jesus. The Koran teaches that Allah has no human counterpart, but that Muhammad is to be honored highly because of the glory of his privlege to receive the final piece of the mystery of human history and existence, which the Koran argues is its own status.
Jesus
The Koran makes note of the historical character of Jesus, whom the Christians believe is one with the Jewish God, Yahweh, and together with the Holy Spirit of God, constitute the trinitarian Godhead.
The Koran does not affirm this theistic view of Jesus Christ. The text does honor him and his teachings, but as a prophet, similar in nature to Moses or Elijah. His teachings of selfless love and transcendence through intimacy with God (Arabic, Allah) are retained though, and Jesus is a special part of the Islamic faith, although the Muslim and Christian ideas of Jesus are certainly incompatible in purely logical terms.
Adam
Adam makes an appearance in the teachings of the Koran, which also maintains that Allah is the creator of this reality and created also the humans, who are the children of Adam, the first man.
Adam is a significant character in the Koran, because the Muslim scriptures teach that Adam is the first prophet, the counterpart to Muhammad who is the final prophet. Adam's theological contribution might be then that God is the Creator (Arabic, Allah), and that humans are subject to the forces of good and evil, and that men are responsible to Allah to do what is good. It is Muhammad, then that finally expounds on that relationship and shows what it is to 'do good.'
Again, because of the theological implications of each religion, it is probably best to understand the Islamic Adam and the the Jewish Adam to be different idealizations of the same mythic character.
