Dragons first came about from Glaurung the Golden, who is said to have been the first fire-dragon bred by Morgoth in the First Age.
But thereafter there was peace for many years, and no open assault from Angband, for Morgoth perceived now that the Orcs unaided were no match for the Noldor; and he sought in his heart for new counsel.
Again after a hundred years Glaurung, the first of the Urulóki, the fire-drakes of the North, issued from Angband's gates by night. He was yet young and scarce half-grown, for long and slow is the life of the dragons, but the Elves fled before him to Ered Wethrin and Dorthonion in dismay; and he defiled the fields of Ard-galen.
The Silmarillion - Chapter 13: Of the Return of the Noldor
There are many speculations of the origins of Dragons, but this question is not asking that. Glaurung is also known as the 'Father of Dragons', or also the 'Father of the Fire-drakes'. Brood is defined as: "a number of young produced or hatched at one time; a family of offspring or young." That would mean that all of the Dragons were descended from Glaurung.
In the front of that fire came Glaurung the golden, father of dragons, in his full might; and in his train were Balrogs, and behind them came the black armies of the Orcs in multitudes such as the Noldor had never before seen or imagined.
The Silmarillion - Chapter 18: Of the Ruin of Beleriand and the Fall of Fingolfin
And but for them Glaurung and his brood would have withered all that was left of the Noldor.
The Silmarillion - Chapter 20: Of the Fifth Battle: Nirnaeth Arnoediad
At last, in the year when Eärendil was seven years old, Morgoth was ready, and he loosed upon Gondolin his Balrogs, and his Orcs, and his wolves; and with them came dragons of the brood of Glaurung, and they were become now many and terrible.
The Silmarillion - Chapter 23: Of Tuor and the Fall of Gondolin
But how-? How did Glaurang reproduce, and subsequently all the Dragons after him? Was it stated anywhere that it was done so asexually or sexually, or through an experiment of Morgoth?