"The Prince That Was Promised", sometimes called "The Prince Who Was Promised" or "The One Who Was Promised", also known as the Lord's Chosen, the Son of Fire and the Warrior of Light, is a mythic figure in the religion of the Lord of Light. According to prophecy, this figure would be reborn "amidst salt and smoke" and pull a flaming sword from the flames, Lightbringer, to combat the coming darkness.
According to Melisandre, a red priestess of the Lord of Light, Stannis Baratheon, Lord of Dragonstone and self-proclaimed King of the Andals and the First Men, is the prophesied Prince. However, after Stannis has fallen and Jon Snow, the Lord Commander of the Night's Watch, is raised from the dead, Melisandre claims Stannis was not the Prince, but "someone must be", suggesting she may now believe it is Jon. She later confirms her belief in Jon as the Prince Who Was Promised to Davos.
Another Red Priestess, established in Volantis, preaches to a group of slaves and freedmen that Daenerys Targaryen is their prophesied savior. The High Priestess of the Red Temple of Volantis is certain that Daenerys is the one who was promised, citing her freeing of slaves and birthing of dragons from stone as evidence. She agrees to send her best priests and preachers throughout Essos, spreading the good word about the Queen. 
In the books
In the A Song of Ice and Fire novels, the Prince That Was Promised is implied to be the reincarnation of a hero known as Azor Ahai. He was a legendary warrior who rose up and defeated the Others - the books' name for the White Walkers - during the Long Night. He wielded a sword of fire, called Lightbringer. There is some debate amongst fans as to the interchangeability of "Azor Ahai" and "The Prince That Was Promised". It seems that Azor Ahai is the name given to the legendary figure, while the Prince That Was Promised is more properly his reincarnation. However, the books do not in fact make explicit the connection between the stories of Azor Ahai and the prophecies of the Prince That Was Promised.
The prophecy about the Prince That Was Promised said he would be born from the line of House Targaryen - Stannis Baratheon does match this description because his grandmother was a Targaryen.
Although Daenerys's gender should raise doubts as to the Red Priestess's identification of her as the Prince That Was Promised, in the books Maester Aemon points out that in High Valyrian (the language most versions of the prophecy were written in), the word corresponding to "Prince" is gender-neutral.
Garth Greenhand · First King · Florian & Jonquil · The Grey King · The Griffin King · Joramun · Lann the Clever
Night King · Night's King & Night's Queen · The Rat Cook · God-on-Earth · The Prince That Was Promised