Arianism was a major heresy in the early Christian church.  It got its name from one of its proponents, the priest Arius who lived in Egypt in the 3rd and 4th centuries.  Arianism was the belief that Jesus was not really God but was, instead, just a regular (if very unusual) human being.

This heresy was believed in by a few Roman emperors and was especially strong among the Germanic tribes.  If it had become the orthodox position, Christianity would be a very different faith.  There would be no Trinity because Jesus would not seen as a divine being.  This would really not be the same faith as the one that we now know as Christianity.

What is the Arian heresy?

This is a good question. The Arian heresy was a christological controversy of the fourth century. There was a bishop named Arius of Alexandria (250-336). He taught that the son was subordinate to the father within Christian doctrine of the Trinity. This caused quite of bit of controversy, since it went against what most people thought.

The upshot was that there was a church wide council in 325 AD - the council of Nicaea, where the church discussed the teachings of Arius. The council, with a lot of help by another bishop, Athanasius, deemed the teachings of Arius a heresy. The council reaffirmed that Father and Son were of the same substance (homousian) and not of a similar substance (homoisousian). The iota in Greek made all the differences. 

Subsequently, the Arian controversy comes up here and there. Modern day Jehovah Witneses can be seen to buy into the Arian teaching by make Jesus subordinate to the Father in being.