On Game of Thrones, we only know of three dragons so far: Daenerys’ dragons. There’s always a possibility that there could be more, but we have no reason to believe that at the moment. The show has indicated that all dragons died out until Daenerys got the three dragon eggs that everyone thought were petrified. But when she stepped into the fire, her three dragons were born.
Dragons are somehow connected to magic. Some think that the red comet brought magic back to the world. But others think it was the birth of the first dragons in 200 years that did it. Still others think the dragons were born in response, somehow, to the coming of the White Walkers.
Valyrians were the first to master dragons and ride them, according to the Game of Thrones wikia. But in the Doom of Valyria, the entire empire of Valyria was destroyed, along with almost all the dragons, in just one day. It’s unclear exactly what caused the doom. The Targaryens were living on a distant island, so they survived.
In fact, House Targaryen had the last surviving dragons in the entire world. They were used to conquer the Seven Kingdoms of Westeros for 300 years. The dragons eventually died and were thought to be extinct everywhere, until Dany came along.
Dany, as a Targaryen, inherited the ability to communicate with and control her dragons (to an extent.)
Daenerys’ dragons are: Drogon, Rhaegal, and Viserion.
Drogon is the dragon that Daenerys has “bonded” with and has ridden. Drogon was named after Khal Drago and was the largest and most aggressive of his siblings. He is black with red markings.
Rhaegal was named after Rhaegar Targaryen and is green with bronze markings. Some think that Jon Snow is going to have a special connection with Rhaegal because he was named after Rhaegar, Jon’s biological father. (In the books, the dragons seem to take after their namesakes to some extent.) In the books, Rhaegal is believed to be much more dangerous than Viserion.
Viserion was named after Viserys Targaryen, Dany’s brother. He’s creamy white with gold markings. Perhaps Tyrion will bond with this dragon?
It’s possible that there are more dragons somewhere in the world, but no one saw any for hundreds of years until Dany’s dragons were hatched. Some fans believe there may be an ice dragon north of the wall — a White Walker version of Dany’s dragons.