Durin's Day is a rare event noted by Dwarves. The first day of the Dwarves' year is the day that begins the last cycle of the Moon, starting with a New Moon, to begin in autumn. When on this day both the Sun and Moon may be seen in the sky together, it is called Durin's Day. Each lunar cycle takes about 29.5 days and autumn in the northern hemisphere runs until about December 21st. The first day of the last new moon of autumn could thus take place any time between about November 22 and December 21.
In The Hobbit, the writing on the map that Gandalf had received from Thráin II mentioned Durin's Day. It predicted that on Durin's Day the last light of the Sun as night fell would reveal the secret door into the Lonely Mountain.