The Connection to Melisandre
Here's the thing: the legend of Azor Ahai and the prophecy of the Prince That Was Promised are technically separate stories — that is to say, no explicit connection between the two figures has been made. Avid readers argue that Azor Ahai is a story, a myth, and the Prince That Was Promised is a prophecy. Even so, Melisandre often speaks of the prophecy and uses its prince interchangeably with Azor Ahai's namesake. What's more, they both wield Lightbringer.
In the show, Melisandre seems absolutely intent on finding the Lord of Light's chosen hero. Up until the end of season five, she believes Stannis Baratheon is the Prince That Was Promised. She continually acts in accordance with these beliefs and even forces Stannis to burn his own daughter alive as a sacrifice to R'hllor. She swears constantly that she sees Stannis's powerful future in the flames. However, he presumably dies at the end of Brienne of Tarth's blade in the season five finale, and Melisandre flees.