The council of Nicea is the name given to the First Ecumenical Council of the Catholic church; the meeting took place in the year 325 CE, beginning either May 20th or June 19th (the actual date is not known.)
The council was called by Constantine the Great, who had converted to Christianity on the occasion of his victory over Maxentius at Milvian Bridge in 312. Constantine had a keen interest in establishing the empire (and emperor) as entities central to the church's doctrine of human salvation, and taking an active role in the affairs of the church was a key part of that strategy. The major dispute that arose in Alexandria during the second decade of the fourth century - later referred to by the church as "the heresy of Arius," which gives one a strong indication of how things ended up - gave Constantine an excellent chance to involve himself in the central politics of the church.
The dispute is almost unintelligible to a modern reader, but constituted a key turning point in the development of the Christian faith. A priest in the Alexandrian church named Arius had begun to preach a version of the holy trinity that asserted that the son and spirit were similar (homoiousious in Greek) and of the same substance. This doctrine was violently opposed first by the city's bishop, Alexander (313-326) and later by his successor Athanasius (236-373), who asserted that the the son and spirit were equal (homoousious) and of the same substance. As with most doctrinal conflicts, it was primarily a matter of semantics and interpretation; nevertheless the trend in Christian literature since has been to lambast Arianism for denying the divinity of the savior and any number of other shockingly impious acts of theological violence, and no doubt opposition at the time was similarly extreme. A local council voted to condemn Arius in 321, but when this failed to put down the argument Constantine saw his chance to get his foot in the door and called a general (or ecumenical) council of the church in an attempt to effect a resolution.
The council was attended by 318 bishops, primarily from the east - the west, which was primarily Latin-speaking and less christianized, was significantly less interested in theological niceties and maneuvering - and from some areas outside the empire. The pope, Sylvester I, did not attend, although he sent a delegation; it is not known whether or not Constantine consulted with Sylvester or ordered the council entirely on his own authority, although it is likely that there was some kind of agreement between the two men. Constantine opened the proceedings, but left the direction of the theological arguments to the bishops. In the end, the council voted to condemn Arius and his teachings with only two dissensions, and Constantine exiled him to Illyria. The bishops also adopted the first formal declaration of their faith, the Nicene Creed.
Not surprisingly, the theological bloodshed didn't end there. The followers of Arius refused to accept the decision, and the abstractness of the officially established doctrine made it less appealing to the masses than some of the tenets of Arianism, which were adopted in part by many more moderate bishops. In 331, Constantine - convinced by ecclesiastical advisors and his sister that Arius had been misrepresented - recalled the exiled bishop, and by the end of his reign had adopted a sort of modified Arianism. His sucessors followed suit rather than compromise the principate's appearance of steadfast faith: Athanasius was exiled and recalled no less than five times under various rulers for various reasons (Julian "the Apostate", for example, saw no better way to damage the church than by bringing the controversy back to the forefront.) Finally, the succession of Theodosius I put an end to the line of Arian emperors, and at the First Council of Constantinople in 381 all forms of Arianism were finally condemned. Despite this fact, Arianism lingered for some time in the army - the emperor Valens had dispatched missionaries to the Goths and converted them to Arianist Christianity, and by that time the Goths formed the backbone of the imperial army. It wasn't until Clovis of the Franks converted to orthodox Christianity in 496 that the theological balance within the ranks shifted and eliminated the "heresy" of Arius once and for all.
Collins, Roger, Early Medieval Europe, 300-1000. St. Martin's Press, New York, 1991.
Leclercq, H., trans. Killeen, Anthony, "First Council of Nicea." Catholic Encyclopedia, http://www.newadvent.org/cathen.
Kiefer, James, "Athanasius, Bishop of Alexandria, theologian." http://www.hillsdale.edu/dept/Phil&Rel/Biography/05/02.html