A glass candle is a candle made of obsidian. Historically they burned, but they have not been reported to burn in recent memory. One green and three black candles were brought to the Citadel a thousand years before the Doom of Valyria. It is unknown if all glass candles are alike, but the ones found at the Citadel are tall and twisted.
The night before an acolyte of the Citadel says his vows, he must stand a vigil in a vault with nothing but the three black candles and no other light permitted. They must spend the night in darkness unless they can light the candle. According to the maesters, it is to show that even with all the knowledge one has acquired, there are still some things that are impossible.