This plot was not explored as deeply in the series as in the book, and it is less clear to the viewer exactly what was going on.
Essentially, Mirri Maz Duur tricked Dany into using Blood Magic to save Drogo's life, at the cost of her baby's life ("Only death may pay for life").
Khal Drogo collapsed, and Mirri told Dany of a blood magic spell she could use which could save Drogo's life. She told Dany that the price for this spell was death, and misled Dany to believe that killing Drogo's horse would be the "death" that the spell required.
Mirri then begins the spell, and many of the Dothraki freak out and start fighting each other; with many of them leaving. While the spell ritual is still happening, Dany experiences painful convulsions.
It felt as if her son had a knife in each hand, as if he were hacking at her to cut his way out.
The remaining Dothraki decide to take her to Mirri, as all the "birthing women" have left. Dany suspects at this point that Mirri has betrayed her, and doesn't want to be taken to her; but is in a lot of pain, and is unable to communicate this, so the Dothraki take her to Mirri.
In Dany's next chapter, she wakes up after a bunch of fever-dreams. When she asks about her child, a handmaid tells her that "the boy... he did not live". The next couple of paragraphs essentially confirms the baby's death:
My son is dead, she thought as Jhiqui left the tent. She had known somehow. She had known since she woke the first time to Jhiqui's tears. No, she had known before she woke. Her dream came back to her, sudden and vivid, and she remembered the tall man with the copper skin and long silver-gold braid, bursting into flame.
She should weep, she knew, yet her eyes were dry as ash. She had wept in her dream, and the tears had turned to steam on her cheeks. All the grief has been burned out of me, she told herself. She felt sad, and yet... she could feel Rhaego receding from her, as if he had never been.
Mirri and Jorah then described what had happened; describing the baby as "monstrous" and that he "had been dead for years"; and Dany understands that Jorah killed her son. Mirri then tells her that her son's death was the price for the spell which saved Drogo's life; not the horse's, only to discover that his life had only been preserved in its near death state, not restored.
After this, Mirri justifies herself to Dany, by saying:
"The stallion who mounts the world will burn no cities now. His khalasar shall trample no nations into dust".
Mirri's motivation for this betrayal is clear. In my opinion she would have no reason to lie about having killed Dany's son. Her motivation was revenge for her temples being burned, and her being raped so many times.
In my mind, there is no doubt that Dany's son died as described. However, the events of what happened between Dany experiencing pain in her womb and waking up are only recounted to us by other people telling Dany; not through a character's POV chapter. Thus, there does exist a slim possibility that the baby could indeed have lived, but I think it extremely unlikely, given that Dany understands so quickly what has happened. She is never in denial or hopes that he lives; she instantly knows and accepts that he is dead. In addition, Mirri's motivation and actions are clear.