Of all the magical beasts in the Lord of the Rings, dragons happen to be the most mysterious — while common in most folkloric legends, these creatures have an unexplained origin in Middle-earth. It is generally believed that the first Dark Lord, Morgoth, created them as his prime minions, but what they were before remains unclear.
There are few named dragons in the narrative, usually depending on their relative importance to the plot, but thousands of them have existed since the dawn of time. The original ones lacked the capacity for flight, which was apparently only evolved much later (consequently making these beasts more dangerous than before.)
The Great Cold Drake
The Great Cold Drake is one of the few ice-based dragons in Middle-earth history and first appears during the war against the Grey Mountain Dwarves. This battle goes on for nearly two decades, where several dragons (known as the Great Worms) destroy King Dain's palaces, with the most powerful one of them murdering the ruler as well as his heir.
The Great Cold Drake is seen by the Dwarves as a near-invincible enemy, much like Smaug several hundred years later, and are forced to flee their homelands to Erebor and the Iron Hills. They thrive for a short period of time, but history inflicts itself on them once again.
Scatha
Scatha is a wingless dragon dating from the time of Morgoth, which suggests that he could be the theoretical ancestor of every dragon born since then (because only he and another unnamed beast survive the Dagor Bragollach and the Nirnaeth Arnoediad). There isn't much revealed of this being's past, aside from the obvious fact that he maintained a massive Dwarven treasure.
Further, some suspect that Scatha is involved in the War Against The Dwarves in the middle Second Age, but this has been proven to be false. The monster is finally killed around two millennia into the Third Age by Fram, who eventually gives rise to the Rohirrim of Rohan. Unfortunately, humans and dwarves both stake claim to the found treasure, leading to more infighting in Northwest Rhovanion.
Gondolin's Fire-Drakes
The Fire-Drakes, better known as Uroloki in Quenya, are supremely mighty, to the point of being able to bring down entire cities by themselves. Speaking of which, this particular specimen is implicated in the destruction of the magnificent city of Gondolin (ruled by King Turgon and Queen Elenwe, grandparents of Earendil, and thereby ancestors of Elrond, Arwen, and Aragorn).
During the Fall of Gondolin, a swarm of Fire-Drakes descend upon the home of the Noldorin Elves, wreaking havoc and causing its inhabitants to flee the city for their lives. Of course, the danger they pose is compounded by the fact that Balrogs are also part of the enemy battalion, leading to the deaths of the rulers as well as some of the most important characters in The Silmarillion, such as Ecthelion of the Fountain.
The Winged Dragons Of The War Of Wrath
The War of Wrath is a massive war fought between the vast military forces of Morgoth against the host sent by the Valar to deliver Middle-earth from its horrifying tyrant. The battles continue for years, with neither side showing weakness until the dark lord begins to sense that a humiliating defeat is on the horizon. In order to shift the tides in his favor, he releases a small "platoon" of flying dragons, the likes of which were completely new to their opponents, and therefore extremely dangerous.
The armies of Eonwe, the herald of the Vala Lord, Manwe, are stunned by these new developments and are temporarily at a loss as to how to deal with this new threat. When it seems that the heroes are about to lose, the half-elven Earendil arrives, just in time, bringing with him all the flying creatures of the world, the greatest of whom are the Eagles. This group manages to successfully defeat and kill all the minor dragons involved, but their leader takes considerably more effort.
Smaug
The main antagonist of The Hobbit, Smaug, is no weakling — his fire-breathing skills are second only to his penchant for gold and violence. Thankfully for the regions around Erebor, he spends most of his time snoring away inside the Lonely Mountain, after forcefully seizing it from the Dwarven ruler, King Thror (Thorin's grandfather). When Gandalf non-consensually involves Bilbo in the adventures, the plan did not include defeating Smaug, because the dragon had been generally considered to be invincible.
However, it's thanks to the little hobbit that he is defeated at the end because Bilbo is the only one who notices the singular point of attack that could potentially kill the beast, a tiny space on his chest (positioned in the neighborhood of his heart.) He conveys this information to Dale, allowing Bard the Bowman to take a nigh impossible shot and bring Smaug down for good.
Glaurung
The firstborn of all the dragons in the world, as well as a major player in the First Age of Arda, is Glaurung. Though wingless, he had the unique capacity to manipulate his victims into madness, as evidenced by his treatment of Turin Turambar (whom he places under a hypnotic charm). The spell is so powerful that it shatters the man's indomitable will and convinces him to change his mission from rescuing Finduilas to going after his sister and mother, who are, in fact, safe in the Sindarin realm of Doriath at this point.
Glaurung repeats this process on Nienor, Turin's sister, going so far as to eradicate all her memories. These two events couple together most unfortunately, leading them to marry each other without realizing that they're related. Several years later, the fates bring all three back together, where the dragon, after being fatally wounded by Turin, unweaves his magic, which consequently leads to the suicide of both Turin and Nienor.
Ancalagon The Black
The most powerful dragon to ever exist since the dawn of time in Middle-earth, and the chief of the Winged Dragons of the War of Wrath, is Ancalagon the Black. His death at the hands of Earendil is legendary because he is killed after considerable effort, although the exact specifics are not mentioned. It's enough to say that his fall utterly destroys the "towers of Thangorodrim," which is a metaphor for the volcanic mountains they are named after.
This means that Ancalagon is probably the largest of his kind, maybe around the same size as the peak itself. As a testament to his power (and that of Sauron and the One Ring), Gandalf informs Frodo that no dragon, "not even Ancalagon the Black could have harmed the One Ring, the Ruling Ring, for that was made by Sauron himself."