The stone dragon is a dragon that Melisandre of Asshai wishes to waken on behalf of King Stannis Baratheon, using king's blood. Melisandre talks of the stone dragon in both the singular and the plural, as "stone dragons".
Only a king's blood can wake the stone dragon. Exactly what Melisandre means when she is referring to the dragon being "stone" is not known, but there apparently is a prophecy that Azor Ahai will "wake dragons out of stone". Thus far there has been no mention of Stannis being in possession of any fossilized dragon eggs, but there are stone carvings of dragons at Dragonstone.
Note that Daenerys Targaryen, born on Dragonstone, was gifted three dragon eggs that were solid stone, fossilized over time, and has hatched them into living dragons in a miraculous ritual under the red comet at the pyre of her husband Khal Drogo, but Melisandre is unaware of this.
A Storm of Swords
On Dragonstone, Melisandre tells Davos Seaworth of a prophecy that when the red star bleeds and the darkness gathers, Azor Ahai shall be born again amidst smoke and salt to wake dragons out of stone, that Dragonstone is the place of smoke and salt, and that Stannis Baratheon is Azor Ahai reborn.
|“||Give me the boy for R'hllor,...and the ancient prophecy shall be fulfilled. Your dragon shall awaken and spread his stony wings. The kingdom shall be yours.||”|
Observing Dragonstone’s stone dragons Davos wonders to himself if the carved dragons were once real dragons, somehow turned to stone by the Valyrians. 
A Dance with Dragons
At the Wall, Melisandre tells Lord Commander Jon Snow that Stannis is the Lord's chosen, destined to lead the fight against the dark. She explains that when the red star bleeds and the darkness gathers, Azor Ahai shall be born again amidst smoke and salt to waken dragons out of stone, and that Dragonstone is the place of smoke and salt. Jon has heard all this before from her, and reminds her that Stannis was not born on Dragonstone, but at Storm's End.
|“||I have seen it in the flames, read of it in ancient prophecy. When the red star bleeds and the darkness gathers, Azor Ahai shall be born again amidst smoke and salt to wake dragons out of stone.||”|
|“||Only a king's blood can wake the stone dragon.||”|
|“||Shireen Baratheon: I had bad dreams. About the dragons. They were coming to eat me.
Maester Cressen: We have talked of this before. The dragons cannot come to life. They are carved of stone, child.
|“||The Lord of Light cherishes the innocent. There is no sacrifice more precious. From his king's blood and his untainted fire, a dragon shall be born.
—Melisandre, to Stannis
|“||Stone dragons who will never wake, I pray.||”|
|“||If the red woman brings them to life, the castle will come crashing down, I am thinking. What kind of dragons are full of rooms and stairs and furniture? And windows. And chimneys. And privy shafts. ||”|
- ↑ 1.0 1.1 1.2 1.3 A Storm of Swords, Chapter 36, Davos IV.
- ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 A Storm of Swords, Chapter 54, Davos V.
- ↑ 3.0 3.1 A Storm of Swords, Chapter 25, Davos III.
- ↑ 4.0 4.1 4.2 A Dance with Dragons, Chapter 49, Jon X.
- ↑ 5.0 5.1 A Clash of Kings, Prologue.
- ↑ A Game of Thrones, Chapter 3, Daenerys I.
- ↑ A Game of Thrones, Chapter 11, Daenerys II.
- ↑ A Game of Thrones, Chapter 72, Daenerys X.
- ↑ A Storm of Swords, Chapter 63, Davos VI.