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
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"