From A Wiki of Ice and Fire
High Valyrian was the language of Valyria and the Valyrian Freehold. It is no longer widely spoken due to the Doom of Valyria, although there are songs and books still sung and read in it. In Westeros, highborn children, specifically the Targaryens, are taught Valyrian as a sign of their noble education. Corrupted dialects known as bastard Valyrian are spoken in the Free Cities and Slaver's Bay.
According to GRRM,
|“||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.||”|
House Targaryen came from Valyria and thus its members can be considered to have High Valyrian names. Such as:
The Valyrians most likely gave names in Valyrian to their dragons, as: "The singers had given them the names of gods: Balerion, Meraxes, Vhagar."  However, not all the House Targaryen dragons have Valyrian names: Queen Alysanne's dragon was named Silverwing. 
As well as Volantis's satellite cities,
The word "dracarys" is translated as meaning "dragonfire."  Obsidian is called dragonglass in the Common Tongue, but frozen fire in High Valyrian. This may be a direct translation of obsidian, the word, but given the word dracarys, it is not likely.  Valonqar is the word for little brother. High Valyrian is the most likely source language for maegi (pronounced differently from 'Maggy') , which means wise. 
The Valyrian writing system, or at least a Valyrian writing system, is described as involving glyphs. It was also probably standard practice to write on scrolls, and not in books. The glyphs can also be inscribed, as on an old Valyrian dragonhorn, which, when sounded, had "every line and letter shimmering with white fire." 
Bastard Valyrian is the languages of the nine Free Cities,. Each of the cities has its own dialect, and each dialect likely has its own separate derived vocabulary. See here for examples of names from each of the Free Cities.
The Free Cities use Valyrian glyphs, as "Arya could not read the name painted on the hull; the words were strange, Myrish, Braavosi, perhaps even High Valyrian."  The Ghiscari certainly use glyphs to write Valyrian.
Valyrian is described as sounding "liquid". 
Astapori Valyrian is described as having a "characteristic growl," influenced by Ghiscari. Also, Syrio Forel of Braavos "had an accent, the lilt of the Free Cities, Braavos perhaps, or Myr." One of the Mummers is described as having a "thick Myrish accent."
Yunkai has its own variations on Valyrian.
- Dany asks: "Missandei, what language will these Yunkai'i speak, Valyrian?"
- "Yes, Your Grace," the child said. "A different dialect than Astapor's, yet close enough to understand. The slavers name themselves the Wise Masters."
Yunkai used to be part of Ghis, and has at least languages tongues which are spoken in the city. Mhysa, Maela, Aelalla, Qathei, and Tato are given as names for mother, but which tongue fits which name is unknown (excepting the first, which is Ghiscari).
Some characters that are familiar with High Valyrian
- Daenerys Targaryen 
- Arya Stark – was taught High Valyrian by Master Luwin but in Braavos the Kindly Man insists that she improve it.
- Septa Saranella – it was she who told Cersei Lannister the meaning of valonqar. 
- Tyrion Lannister – learned to read High Valyrian on his maester’s knee.
- Ser Barristan Selmy - has some High Valyrian, though not as much as Dany 
- Melisandre - known to pray in High Valyrian. 
- Samwell Tarly – only has a little High Valyrian 
- Gerris Drinkwater - speaks a halting approximation of High Valyrian 
- Quentyn Martell - can read and write High Valyrian but has little practice speaking it. 
- Aegon Targaryen - is fluent in High Valyrian 
- Haldon Halfmaester
- The Tattered Prince - he says and now we ride to his men in a classic High Valyrian. 
- The men of the Windblown - the closest thing they have to a company tongue is classic High Valyrian. 
- Sweets – is fluent in High Valyrian. 
Valyrian languages on Wikipedia.
References and Notes
- ↑ A Game of Thrones p.87
- ↑ A Storm of Swords p.676
- ↑ [ http://www.westeros.org/Citadel/SSM/Entry/1250/ So Spake Martin]
- ↑ A Game of Thrones p.87
- ↑ A Game of Thrones pages 832-5
- ↑ A Clash of Kings p.104
- ↑ A Storm of Swords p.752
- ↑ A Storm of Swords p.453
- ↑ A Clash of Kings p.25
- ↑ A Game of Thrones p.122
- ↑ A Storm of Swords p.453
- ↑ A Storm of Swords p.X
- ↑ A Clash of Kings p.520
- ↑ A Storm of Swords p.308
- ↑ A Feast for Crows p.95
- ↑ A Feast for Crows p.507
- ↑ A Storm of Swords p.93
- ↑ A Storm of Swords p.885
- ↑ A Feast for Crows p.584
- ↑ A Feast for Crows p.544
- ↑ A Game of Thronesp.798
- ↑ A Game of Thrones p.33
- ↑ A Clash of Kings p.581
- ↑ A Feast for Crows p.277
- ↑ A Game of Thrones p.24
- ↑ A Storm of Swords p.359
- ↑ A Game of Thrones p.101
- ↑ A Game of Thrones p.721
- ↑ A Storm of Swords p.262
- ↑ A Clash of Kings p.429
- ↑ A Storm of Swords p.258
- ↑ A Game of Thrones p.244
- ↑ A Game of Thrones p.441
- ↑ A Storm of Swords p.476
- ↑ A Storm of Swords p.487
- ↑ 36.0 36.1 A Storm of Swords, Chapter 23, Daenerys.
- ↑ A Feast for Crows, Chapter 39, Cersei.
- ↑ A Clash of Kings, Chapter 10, Davos, p 146.
- ↑ A Feast for Crows, Chapter 26, Samwell.
- ↑ 40.0 40.1 40.2 A Dance with Dragons, Chapter 6, The Merchant's Man (Quentyn I).
- ↑ 41.0 41.1 A Dance with Dragons, Chapter 25, The Windblown (Quentyn II).
- ↑ A Dance with Dragons, Chapter 47, Tyrion.
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