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.
Unlike Tolkien, who created entire languages, including grammar, syntax, and tenses. Martin has created only a few words from each language, his languages are not conveyed in any great detail. Instead, the tongue in which they are speaking is noted but rendered in English, with added characteristics and flavour portrayed through out the text.
"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.
However for the release of 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".
Languages of Westeros
Although there are many spoken languages on Westeros, only two are native to it: the Old Tongue, the language of the First Men that dominated westeros thousands of years and the Andal language, today known as the Common Tongue, which was introduced during the andal invasion/migration and superseded the old tongue since. Another major migration to westeros was the made by the Rhoynar, however they have assimilated and their own language has disappeared since.
Today, the Old Tongue is still remembered by the folks in the far north, while most people within the Seven Kingdoms, know no other language than the Common Tongue. Although there are minor variations to it from place to place, Dornishmen can communicate with Northmen without difficulty. Foreign tongues are taught in the Citadel; otherwise, otherwise, they are brought from other lands by immigrants, merchants, sellswords, and the like.
Languages 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.
- Ghiscari - is largely extinct tongue, spoken five thousand years ago by the Ghiscari people of ancient Ghis, destroyed by the Valyrian Freehold. Ghiscari was supplemented by a bastardised vestion of High Valyrian, though it is still spoken by some and engraved Ghiscari glyphs can be found on various objects of old.
- 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, said to sing in the True Tongue.
- The Others' Language, described as sounding like cracking of ice on a winter lake. In the Game of Thrones TV show, It was named "Skroth".
- Dothraki wiki, on languages