When is it introduced?
The first time we hear about this prophecy is with the introduction of everyone's favorite princess-burning priestess in season two, Melisandre. She worships the Lord of Light, and followers of the Lord of Light believe that a champion of their god called the Son of Fire or the Prince That Was Promised (also called Azor Ahai) is destined to rise up and fight against the darkness that threatens the world. Melisandre believes that this "darkness" will come in the form of the White Walkers. She tells Stannis in the third season that the War of the Five Kings is nothing compared to the real war at the Wall.
Melisandre has all her chips on Stannis Baratheon being the prince in question for a long time and staunchly believes he will defeat all enemies standing in the way of him becoming the ruler of the Seven Kingdoms. Her fervent faith in him motivates him to do basically all the stuff he does.
However, Stannis's chances of ruling the Seven Kingdoms and fulfilling the prophecy are pretty much destroyed by the end of season five, when his depleted army is decimated by the Boltons. At the beginning of season six, Melisandre believes she's lost her ability to interpret the prophecy, but her resurrection of Jon Snow (woot!) reignites her faith in her Lord and that the Prince That Was Promised wasn't Stannis, but Jon himself.
What does it mean for the future?
After Jon's badassery at Hardhome (and the implications that battle makes about what the White Walkers can do), the idea that Jon Snow could lead the people of Westeros against the darkness in the world is a pretty safe bet — if there's any such thing as a safe bet on this show. Melisandre's current faith in Jon is reminiscent of the faith she had in Stannis, but since Jon has banished her to the South after discovering she was the one who burned Shireen, who knows how their relationship will play out.
NOTE: Jon isn't the only one who might be the Prince That Was Promised — there are quite a few who think the prophecy applies to Daenerys, instead.
