“Moon-letters are rune-letters, but you cannot see them,” said Elrond, “not when you look straight at them. They can only be seen when the moon shines behind them, and what is more, with the more cunning sort it must be a moon of the same shape and season as the day when they were written. . . . These must have been written on a midsummer’s eve in a crescent moon, a long while ago.”

The moon-letters included instructions for discovering the entrance to the dwarf city of Erebor, underneath the Lonely Mountain, which was in the possession of the evil dragon Smaug. These instructions taught that . . . 

“The setting sun with the last light of Durin’s Day will shine upon the key-hole.” 

“Then what is Durin’s Day?” asked Elrond. 

“The first day of the dwarves’ New Year,” said Thorin, “is as all should know the first day of the last moon of Autumn on the threshold of Winter. We still call it Durin’s Day when the last moon of Autumn and the sun are in the sky together.” 

This is a clear reference to the time of the New Moon, which is traditionally reckoned to be the first appearance of the crescent Moon in the evening sky after being invisibly lost in the Sun’s glare at the end of the previous lunar month. During the time of the New Moon, the Moon is very close to the Sun and rises around the same time as the Sun. Thus, the Moon crosses the sky with the Sun and is seen only in the evening after sunset, when the Moon itself is very near to setting. 

“The first day of the last moon of Autumn” refers to the New Moon before the winter solstice, December 21. Thus, Durin’s Day is a “moveable feast,” like Easter, and could fall any time between late November and December 20, depending on the difference between the cycles of the Sun and Moon in any given year. The calendar reckoning needed to determine this exact date in advance requires a detailed understanding of astronomy. Thorin laments that he and his dwarves did not possess such knowledge: 

“But this will not help us much, I fear, for it passes our skill in these days to guess when such a time will come again.” 

I wonder if Tolkien meant this as his own lament of modern culture, where such information “passes the skill” of most people, in his time and still in ours.

After much adventuring, Bilbo and the dwarves finally arrived along the side of the Lonely Mountain, in the proper season of Durin’s Day, as Thorin remarks that “tomorrow begins the last week of Autumn.” As they found themselves at the entrance to Erebor, it was Bilbo who providentially discovered the circumstances of Durin’s Day, as well as the clues to open the door:

As the sun turned west there was a gleam of yellow upon its far roof, as if the light caught the last pale leaves. Soon he saw the orange ball of the sun sinking towards the level of his eyes. He went to the opening and there pale and faint was a thin new moon above the rim of Earth. 

The sun sank lower and lower, and their hopes fell. It sank into a belt of reddened cloud and disappeared. The dwarves groaned, but still Bilbo stood almost without moving. The little moon was dipping to the horizon. Evening was coming on. Then suddenly when their hope was lowest a red ray of the sun escaped like a finger through a rent in the cloud. A gleam of light came straight through the opening into the bay and fell on the smooth rock-face. . . .