High Valyrian is the language of the old Valyrian Freehold which was located on the eastern continent of Essos.
Much of Essos was once dominated by the Valyrians for thousands of years, stretching from the Free Cities in the west, to Slaver's Bay in the east. The Valyrians forced the peoples they subjugated to speak in (or at least be able to converse in) their language.
After the Doom of Valyria 400 years before the War of the Five Kings, "High Valyrian" (as it became known) ceased to be a living language, instead being used as a lore-language by scholars throughout both Essos and Westeros.
Lexicon
Dracarys - ("drah-KAH-ris")[1] The High Valyrian word for "dragonfire". Daenerys says "Dracarys" to Drogon, the young dragon, to encourage him to breathe fire.[2] Daenerys Targaryen teaches her newly hatched dragons to breathe fire on command when she says dracarys.[3]
Valar morghulis - (va-lar mor-gu-lis)[4] "all men must die", in the sense of "all men must (eventually) die (sooner or later)". A common saying in Braavos, particularly among the guild of assassins known as the Faceless Men.[5]
Low Valyrian
Without the central influence of the Valyrian empire, the speech of their descendants and former colonies mutated into a romance language known as "Low Valyrian", or "Bastard Valyrian". Low Valyrian is not one language, so much as a family of diverse dialects well on their way to becoming separate languages - so much so that even those who speak one might not be able to speak another, and without mutual intelligibility, it could be argued that they have truly become separate languages.
Each of the Nine Free Cities has its own dialect/language of Low Valyrian. These include Braavosi, Lorathi, Lysene, Myrish, Norvoshi, Pentoshi, Qohorik, Tyroshi, and Volantene. Slaver's Bay also has its own dialect/language of Low Valyrian, making for a total of ten different branches.
The Low Valyrian of Slaver's Bay is somewhat influenced by the old language of Ghiscar, but owes more of its descent to High Valyrian than the old local languages.
In the books
As a dead language of a fallen empire used by scholars and educated noblemen throughout a medieval world, High Valyrian is basically their world's equivalent of Latin. Indeed, author George R.R. Martin has said that the old Valyrian Freehold is basically his fantasy world's equivalent of the Roman Republic. Most of the people in the eastern continent of Essos speak one of the romance languages that derive from High Valyrian, including all of the Free Cities and all of Slaver's Bay.
Tyrion Lannister learned High Valyrian from his tutors growing up, as apparently many of the noble-born of Westeros do. Thus even though the Valyrian Freehold never extended to Westeros, many of the better-educated characters are capable of understanding it. Numerous times throughout the books, Tyrion or other characters are observed reading books written in High Valyrian. Even Winterfell had several rare books written in High Valyrian in its collection.
As for Low Valyrian, Tyrion could speak some Braavosi reasonably well, knew a smattering of Myrish, and a few phrases of Tyroshi. Tyrion apparently could not speak Pentoshi. Tyrion himself says that Low Valyrian is "not so much a dialect as nine dialects on the way to becoming separate tongues". However, the fact that Tyrion knows High Valyrian and some dialects like Braavosi, but cannot understand others, lends a large amount of evidence to the position that they have reached the point where they are not mutually intelligible and are truly separate languages.
In contrast, the Low Valyrian of Slaver's Bay, which is influenced by the old Ghiscari language, is more uniform. It is stated that while the Low Valyrian of Yunkai is a different dialect than that of Astapor, they are still mutually intelligible. However, a great multitude of other languages are spoken in large numbers in Slaver's Bay, due to the large influx of slaves from many distant lands.
Valyrian is described as sounding "liquid". The Valyrian writing system is said to involve glyphs.
House Targaryen, a noble family of the Valyrian Freehold living on their most distant outpost on Dragonstone island, survived the Doom along with their Dragons. Presumably, the Targaryens of Dragonstone spoke High Valyrian as well. After the Targaryens conquered Westeros three hundred years ago, they conversed with their subjects in the Common Tongue of the Andals which was spoken throughout the continent. It is not clear if the Targaryens stopped speaking High Valyrian entirely, or continued to teach it to their children. Certainly, the Targaryens were proud of their descent from Old Valyria, and the children of powerful noble Houses like Tyrion Lannister learned High Valyrian, so it stands to reason that even Rhaegar Targaryen's generation could speak High Valyrian.
Daenerys Targaryen knows High Valyrian, though how she learned it in exile isn't explained. Daenerys has been observed in the TV series saying words of High Valyrian, like "dracarys". Daenerys is also stated to know the (Low) Valyrian of the Free Cities, because she grew up there, though which variants she knows are not clear. When Daenerys responds to a merchant in Vaes Dothrak speaking in "Valyrian", the variant she replies in makes him think she is from Tyrosh, so she seems to be able to speak Tyroshi. Daenerys may be reasonably familiar with Braavosi and Pentoshi, because those were the Free Cities that she spent the longest time in. She also briefly stayed at various times in Myr, Qohor, Volantis, and Lys, so she may have some familiarity with those variants as well. However, Daenerys cannot understand the Low Valyrian spoken in Slaver's Bay, which is very different from those spoken in the Free Cities.