The quarrel began over a practice. Monks on Mount Athos sat in stillness, controlled the breath, fixed the gaze, and repeated a short prayer continuously, and some of them reported seeing a light which they said was uncreated, the same light the disciples saw on the mountain at the Transfiguration. Barlaam, a learned Calabrian Greek, found the physical technique absurd and the claim impossible: God is unknowable in his essence, and a created being cannot see anything uncreated, so what the monks saw was either imagination or something made.
Palamas, himself a monk of Athos, answered in a series of treatises defending the holy hesychasts. His argument concedes the premise and denies the conclusion. God's essence is indeed wholly unknowable and always will be. But God is not only his essence: he acts, and his actions, the energies, are God himself in operation rather than created effects. The light on Tabor is one of them. So the monks are seeing God, truly, without seeing the essence, and the unknowability the philosophers insist on is preserved exactly.
The dispute ran for two decades and was entangled with a civil war in which the two sides had political patrons. Councils at Constantinople in 1341, 1347 and 1351 upheld Palamas, Barlaam left for Italy and the Latin church, and Palamas was made archbishop of Thessalonica and canonised within a decade of his death.