When is it introduced?
When Daenerys becomes pregnant by Khal Drogo in season one, it's more than just good news for the happy couple. The dosh khaleen in Vaes Dothrak proclaim that their son will be much more than Dothraki royalty — he will be the Stallion Who Mounts the World, the khal prophesized to bring all the khalasars together into one people. (Dany recounts this prophecy in season six, just before she burns the crap out of those mean khals.)
Tragically, by the end of season one, Daenerys's unborn son is dead, as a result of the witch Mirri Maz Duur's dark magic to "save" Khal Drogo, who ends up comatose. Mirri Maz Duur makes a sort of bitter prophecy of her own, telling Daenerys that her husband would return to her "when the sun rises in the west and sets in the east." Daenerys ends up mercy killing Drogo because, obviously, those words don't suggest that her husband had much hope.
What does it mean for the future?
Dany brings up the topic of bearing children to Khal Moro in season six when he threatens to rape her and get her pregnant, when she claims that she will never bear him or anyone else any sons. While Dany is certainly fierce enough to simply refuse to bear anyone else children after losing her beloved husband, she could also have meant that Mirri Maz Duur's magic did too much physical damage to her womb, which would imply that Daenerys won't be bearing any world-mounting children anytime soon.
However, in the very same episode that her son dies, Daenerys still becomes a mother, in a way: The Mother of Dragons.
