Short answers, pulled from the story.
Grace in Christianity is God's favor and a share in the divine life of God, described as a spontaneous gift that is generous, free, and undeserved and that cannot be earned. The Greek word charis means that which brings delight, joy, happiness, or good fortune.
Sanctifying grace, also called habitual grace, is the divine life infused into a person's soul once they are justified, a stable supernatural disposition that perfects the soul. Actual grace refers to God's punctual interventions that produce, maintain, or increase sanctifying grace. The distinction was drawn by St. Thomas Aquinas in his Summa Theologiae.
The Council of Trent in 1547 established Catholic teaching that justification and sanctification are part of the same process and that the sacraments are principal means of grace. Martin Luther taught that salvation comes by faith alone and grace alone, with no room for merit, after posting his Ninety-five Theses in Wittenberg in 1517.
Pelagius, an ascetic said to have come from Britain, blamed moral laxity in Rome on Augustine's theology of divine grace and insisted humans had free will to choose good or evil. The Council of Carthage repudiated Pelagianism in 418, largely at Augustine of Hippo's insistence.
Calvinism teaches divine monergism, double-predestination, and irresistible grace, holding that God alone authors every stage of salvation. Arminianism, formulated by Jacobus Arminius in the early seventeenth century, teaches that prevenient grace comes to all people and can be freely accepted or freely denied.
The Eastern Orthodox Church identifies grace with the uncreated Energies of God, not a created substance that can be treated like a commodity. It teaches that the human will must cooperate with divine grace through synergism so that a person may become deified in a process called theosis.