One night, sometime before Harry Potter was born, two people listened intently as a prophecy was revealed concerning the fate of Voldemort, and a yet unborn child. The first listener only heard a portion of the prophecy, learning merely that the time of birth was approaching of a child who could defeat Voldemort. The second listener, however, learned the entire contents of the prophecy.
"The one with the power to vanquish the Dark Lord approaches... Born to those who have thrice defied him, born as the seventh month dies... And the Dark Lord will mark him as his equal, but he will have power the Dark Lord knows not... And either must die at the hand of the other for neither can live while the other survives."
It was these words, spoken through Sibyll Trelawney to Albus Dumbledore, that set in motion several important events:
1. Part of the prophecy was heard by a Voldemort supporter (Severus Snape). Voldemort learns only that a boy born to parents who have defied him three times will be able to vanquish him.
2. Voldemort apparently decides to go after Harry Potter, although Neville, a pureblood, would have qualified under the prophecy as well.
3. The Potters learn through a spy that Voldemort intends to go after them. They give their location to a secret keeper, Peter Pettigrew, and go into hiding.
4. Peter Pettigrew reveals the Potter's location.
5. On October 31st, Voldemort comes to the Potter household. He kills James on the spot, but offers to let Lily step aside so he can kill Harry. Lily refuses and dies protecting her son, putting forward an ancient magic of protection that Voldemort doesn't understand.
6. Voldemort turns his wand to Harry, and casts a spell to kill him. The spell rebounds (thanks to Lily's protection), almost completely destroying Voldemort.
Harry did not learn all of this until his fifth year at Hogwarts. Voldemort, through the use of dreams, made Harry think Sirius had been captured in the Department of Mysteries. It was all a ruse, however, to get Harry to a room where all the records of prophecies were stored. Only Harry and Voldemort could pick up the record from the shelves. In the ensuing battle, the prophecy was destroyed, once again denying Voldemort full knowledge of the prophecy.
During Harry's sixth year, Dumbledore explained to Harry exactly what the prophecy means. Prophecies do not have to be fulfilled. But, because Voldemort believed in the prophecy, he has caused portions of it to move forward. He became obsessed with the idea that someone could destroy him. And so he went after Harry, and in the end gave Harry some of his powers which can help Harry bring Voldemort down. Harry also has one other very important power and that is the ability to love. This is the power that Harry has that Voldemort does not know or truly understand.
The prophecy only matters in the end because Voldemort made it matter and has made at least some of it come true so far.