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.