Fathers of the Dwarves
The Seven Fathers of the Dwarves were the first of their race.
The Vala Aulë created the Dwarves because he was impatient for the arising of the Children of Ilúvatar (Elves and Men) and he wished for children to love and instruct. It was the period known as the Sleep of Yavanna when Middle-earth was dark and silent and was roamed by monsters from Utumno. For this, Aulë made them strong and resistant and able to endure hardships; but as he had only a vague impression of the Children of Ilúvatar, his creations were structured differently, shorter and stunted.
He created seven Dwarves, and was teaching them the language he had devised for them (Khuzdul), but Aulë was not Ilúvatar who had the Flame Imperishable and his children were dumb, able to move and speak only if he wished so, and would remain motionless whenever Aule would think elsewhere.
Ilúvatar confronted him for his impatience. Aulë offered his creations to Ilúvatar and was ready to break them in repentance. But Ilúvatar accepted his offer and gave them life of their own, and the Dwarves started cowering and pleading for mercy despite Aule's will.
However, the Fathers of the Dwarves had to wait until the Elves first arrived, and Aulë laid them to rest in various places in Middle-earth. The eldest of all, Durin, "lay alone" at Mount Gundabad in the north of the Misty Mountains. He later founded the line of the Longbeards (or Sigin-tarâg in Khuzdul), the Dwarves which were most friendly to the Elves and Men, mostly referred to as Durin's Folk. His city was Khazad-dûm.
Two others were laid to rest in the Ered Luin or Blue Mountains, and they founded the lines of the Broadbeams and the Firebeards who later lived in Nogrod and Belegost. The other four Fathers of the Dwarves were laid to rest in two pairs in Rhûn; lands at least as far east of Mount Gundabad as it lay east of the Blue Mountains. They founded the lines of the Ironfists, Stiffbeards, Blacklocks, and Stonefoots.
It was said that the Seven Fathers would return to life after generations among their own folk. This did not mean reincarnation, however. Rather, the bodies of the Fathers would be preserved, and their spirits would return to the bodies and live once more. Durin was known to return to life six more times. 
Of the Fathers of the Dwarves, only Durin is said to have "lain alone".
Other versions of the legendarium
The reference to Durin being alone can be interpreted as referring to the fact that, in older versions of the story, Durin alone had no female companion (references are made by Tolkien to the "Thirteen Dwarves" created by Aulë, Durin and the six pairs[source?]) but later it meant that Durin was indeed laid down to rest alone while the other Fathers were laid to rest as three pairs. In that same conception, Tolkien considered that there were other Dwarves made that were put to sleep beside their Fathers; Durin would be again an exception, as he created his Folk by gathering Dwarves from other clans during his wanders.
In the Lost Tales there were only two groups of Dwarves, the Nauglath of Nogrod under their king Naugladur and the Indrafangs, the Longbeards of Belegost under their king Bodruith. They live south of Artanor (Doriath).
According to the Tolkien Encyclopedia, Aulë's creation of the Dwarves is sub-creation which aims to honor the wider Creation of God/Eru, a concept expressed in Mythopoeia; "and may indicate anxieties about the independent value of art."
- J.R.R. Tolkien, Christopher Tolkien (ed.), The Silmarillion, "Quenta Silmarillion: Of Aulë and Yavanna"
- J.R.R. Tolkien, Christopher Tolkien (ed.), The Peoples of Middle-earth, "Of Dwarves and Men"
- J.R.R. Tolkien, The Lord of the Rings, Appendix A, "Durin's Folk"
- J.R.R. Tolkien, Carl F. Hostetter (ed.), The Nature of Middle-earth, "Part Two. Body, Mind and Spirit: XV. Elvish Reincarnation"p. 264-265
- J.R.R. Tolkien, Christopher Tolkien (ed.), The Peoples of Middle-earth, "X. Of Dwarves and Men", "Notes", #24
- J.R.R. Tolkien, Christopher Tolkien (ed.) 1984: The History of Middle-earth, vol. 2, The Book of Lost Tales, Part Two, pp. 223, 230, 233.
- J.R.R. Tolkien, Christopher Tolkien (ed.)1984: The History of Middle-earth, vol. 2, The Book of Lost Tales, part two, p. 225.
- Michael D.C. Drout, ed., J.R.R. Tolkien Encyclopedia: Scholarship and Critical Assessment, p. 134