Durin's Day was a yearly event noted by the Dwarves, "when the last moon of Autumn and the sun are in the sky together" (i.e., until sunset) on first day of the dwarves' New Year, which was "the first day of the last moon of Autumn on the threshold of Winter." Since each lunar cycle takes about 29.5 days and autumn (by the American definiton of winter) in the northern hemisphere runs until about December 21st, the first day of the last new moon of autumn could take place any time between about November 22 and December 21. However it is more likely that Tolkien would have used the looser British definiton of winter which starts at the beginning of December which means that Durin's day could fall anywhere from Mid-October to late November.
In The Hobbit, the writing on the map that Gandalf had received from Thráin II, as read by Elrond, instructed, "Stand by the grey stone when the thrush knocks, and the setting sun with the last light of Durin's Day will shine upon the key-hole" (of the secret side-door into the Lonely Mountain). And later on in the story, this is what happened.