From A Wiki of Ice and Fire
The Balerion, with Viserion
atop the cog's figurehead - by Tomasz Jedruszek. © Fantasy Flight Games
aboard the Balerion
with her three dragons - by Mark Evans. © Fantasy Flight Games
Magister Illyrio Mopatis sent the Saduleon, Joso's Prank and the Summer Sun to Qarth to bring Daenerys Targaryen back to Pentos in preparation for her return to Westeros. Daenerys accepted, but insisted in renaming the ships as Vhagar, Meraxes and Balerion, in homage of the three powerful dragons of Aegon the Conqueror.
Balerion is the great cog previously known as Saduleon; Meraxes is the galley previously known as Joso's Prank; and by implication, Vhagar is the galley formerly named Summer Sun.
When Daenerys took Meereen, she found herself in want of the wood and masts of the Balerion and the Meraxes for siege weapons and of the masts for battering rams.