Most dragons, and Valyrians, perished during the Doom of Valyria. Those that survived were those residing on Dragonstone: House Targaryen. Originally a minor house of the Valyrian Freehold, the Targaryens suddenly found themselves the only survivors of their race and masters of the three remaining dragons:
- Balerion the Black Dread
The last dragon of House Targaryen (and supposedly in the world) died approximately 150 years before the events of A Song of Ice and Fire. Most of the dragons died in combat (with the majority coming from the Targaryen civil war called the Dance of Dragons). Each successive generation of dragons was smaller, sicklier, and shorter-lived than the previous.
It would also be good to point out that a lot of dragons died during the Dance of Dragons (Not the 5th book of the series — which was called A Dance With Dragons– , but the civil war that happened during the Targaryen rule). This civil war was on GRRM’s novella The Princess and The Queen, or, The Blacks and The Greens, part of the Anthology Dangerous Women.
The dragons that died during the civil war were:
Vhagar, Arrax, Caraxes, Vermithor, Vermax, Tyraxes, Dreamfyre, Grey Ghost, Tessarion, Syrax, Sunfyre, Seasmoke, Morghul, Moondancer, Meleys, Stormcloud and Shrykos.
There weren’t many of them to begin with, and as you can see, quite a lot perished on that civil war (129 – 131 AC)