Durin's Day is a rare event noted by Dwarves.
The new year of their calendar was the last cycle of the Moon to begin in autumn, or in Thorin's words "the first day of the last moon of Autumn on the threshold of Winter". When on this day both the Sun and Moon may be seen in the sky together, it is called Durin's Day.
Unlike The Lord of the Rings Tolkien left no precise timetable for the events of The Hobbit; however according to Andreas Möhn, the Durin's Day of T.A. 2941 occurred on 22 October of the Shire Calendar. This is based in the assumption that the orbital period of the moon, as described in LotR, was about 20 minutes shorter/faster than today.