The Prophecy was a prediction made by the Oracle and, as told to Neo by Morpheus, states the coming of The One and that it will herald the destruction of the Matrix and the freedom of humanity from their oppression by the Machines. Once The One enters the Source, he will have the power to destroy the Matrix.
Morpheus learns of the Prophecy early in his life from a mysterious woman known as the Oracle, who in reality is a program. Morpheus becomes consumed in his quest to find The One, leaving his lover Niobe to seek other company, and confusing and confounding some Zion officials and hovercraft operatives who do not share his belief in the Prophecy.
Actually, the Prophecy is an invention of the machines, told to Zion operatives by the Oracle; the Prophecy is designed to manipulate the humans into leading The One to meet the Architect and not the Source itself. The One learns the truth and is forced into reloading the Matrix under the consequence that the Matrix will crash and all of humanity will die if The One does not comply.
Neo, once enlightened and manifest in his powers as The One, finds himself confused as to what to do to save humanity six months after his transformation. The Oracle points him to find The Keymaker, a program that will help him locate the Source. While Neo successfully enters the door to the Source, he meets the Architect. The Architect explains that the Prophecy was a control to direct The One into saving a portion of humanity to rebuild Zion (after the Machines destroy the current city) as well as all humans connected to the Matrix. If The One defies the Architect and returns to the Matrix, the Architect claimed that all humanity would die once the Matrix crashes and once the last human in Zion is killed.
Neo, in his love for Trinity, changes the equation and stops the cycle. He defies the Architect and returns to the Matrix to save Trinity from death. When this event happened, Neo initially believes the Prophecy to be false until the Oracle later sets him again to meet the Source--not the Architect, as before, but The Source located in the Machine City. When Neo barters a peace with the Deus Ex Machina on the condition of the destruction of Smith, the Prophecy is fulfilled, and the humans and Machine enter a period of peace.