Montanism was a heresy that was started by a man named Montanus who lived in the Roman Empire about 170 AD. Montanus lived in Phrygia (modern Turkey). He preached that God might appear at any time and talk to pretty much anyone, according to the Christian historian Eusebius (who hated him). Montanus claimed that the ability to see Jesus and to be possessed by the Holy Spirit had not ended with the disciples and Paul, but was an ongoing thing which might happen to any Christian at any time, if he or she was holy enough. Montanus said that he, himself, was holy enough and that he had been possessed by a spirit which had given him new information about how the Church was supposed to work. This fits in well with the general Greek and Roman tradition of holy men, and so did Montanus by all accounts: he had long hair and bare feet and looked like other Eastern prophets. Apparently he was very convincing in person, especially to women, who often became possessed themselves. His followers began to collect his sayings in a book called Revelations.
One of the things which God revealed to Montanus was that the world was going to end very soon, maybe in 200 AD. There was a great plague sweeping the Empire at this time, which made him more convincing.
But the Church thought that the idea of just anybody being able to hear God and switch everything around was very disturbing. In response to Montanism, the Christian bishops developed the concept of possession by demons. The bishops pointed out to their congregations that Jesus had warned against false prophets. In the end, Montanus was excommunicated as well, but by then he had his own followers who stuck with him.
In fact, Montanus not only attracted a lot of people to his cause — people thought it was exciting that God might enter into them at any time — but he kept them: there were Montanists well into the 500s A.D. (despite the world not ending on schedule). Tertullian, for example, was a Montanist several decades later, writing about 200 A.D. In general, Montanism was better accepted the further from Phrygia you got, where the bishops didn't feel so threatened by it. The Bishop of Rome at this time saw nothing wrong with it, and Tertullian saw no contradiction between Montanism and orthodox Christianity.