"I don't have a whole imaginary language in my desk here, the way Tolkien did."
Tolkien was a philologist, and an Oxford don, and could spend decades laboriously inventing Elvish in all its detail. I, alas, am only a hardworking SF and fantasy novel, and I don't have his gift for languages. That is to say, I have not actually created a Valyrian language. The best I could do was try to sketch in each of the chief tongues of my imaginary world in broad strokes, and give them each their characteristic sounds and spellings.
Throughout A Song of Ice and Fire, numerous cultural and regional dialects are discussed. These dialects form the world's spoken tongue, and help define it's people.
Many tongues are spoken on Westeros, but only two are native to it: the Old Tongue, which was spoken before the Andal invasion, and the Common Tongue, which followed it. It may be possible that the Rhoynar brought their own language with them, but this has since disappeared. Foreign tongues are (probably) taught only in the Citadel; otherwise, they are brought from other lands by immigrants, merchants, sellswords, and the like.
Across the narrow sea
- Dothraki language, is the language of the Dothraki, the indigenous inhabitants of the Dothraki Sea in the series A Song of Ice and Fire written by George R. R. Martin.
- High Valyrian, was the language the Valyria empire, before the Doom came and it fell apart.
- Low Valyrian, Bastard Valyrian, as it is also called, is the language of the nine Free Cities and the Slaver Cities, Each of the cities has it's own dialect, and each dialect likely has it's own separate derived vocabulary.
- Tyroshi, Little is known about Tyroshi Low Valyrian, except concerning its pronunciation.
- Ghiscari - The Ghiscari tongue is largely forgotten, thou it's still spoken by some. the slave cities, remnants of old Ghiscar, speak the High Valyrian of their conquerors or at least a bastard version of it.
- Language of Asshai, The people of Asshai, are well versed in witchcraft and wizardry, and have a language of their own, used in their spells (as did the Valyrians).
- Lhazareen - is described as singsong 
- Summer Tongue, Summer Islands have their own language.
- The Trade Tongue - The trade tongue is often spoken, a coarse argot that has developed using words from a dozen languages (many of them insults) and hand gestures 
- Other Languages
- The Children of the Forest
- The Others' Language, "The Other said something in a language that Will did not know: his voice was like the cracking of ice on a winter lake, and the words were mocking."
Created for HBO's Game of Thrones, a actual spoken Dothraki language has been developed. This language takes cues from George R. R. Martin's few bits of published words, and expands greatly upon it. A website, dothraki.org, has been created to help teach and spread the constructed tongue. Film critic and indie director Lee Demarbre has called the language the "new Klingon".
- Dothraki wiki, on languages