I heard 'Durin' many times in the hobbit movie but don't understand if it's a person or some specific day or what?
Durin was one of the oldest of the seven fathers of the dwarves. When they talk about Durin they are usually talking about the line of Durin meaning that the said dwarves are related to him.
King Durin I of Khazad-dûm (Years of the Trees? - First Age), better known as Durin the Deathless, was the eldest of the Seven Fathers of the Dwarves, first created by Aulë the Valar. Durin was set to sleep alone beneath the mountains of Middle-earth until the Elves were born. The name Durin, like other names of Tolkien's Dwarves, was taken from Old Norse. In Tolkien's fiction, Dalish (the language of Dale) had the same relationship to Rohirric and Westron as Old Norse has to English, so since Westron was rendered in English, Dalish was rendered in Old Norse. The Dwarvish names were in Dalish (the language of Dale), which therefore was rendered in Old Norse in The Hobbit and The Lord of the Rings.
Here is a depiction of the lineage of Durin.
As mentioned Durin's day is an important part of the story as well.
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 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.
The Hobbit, Chapter III: "A Short Rest"