The Prophecy

The prophecy made by Sybill Trelawney to Albus Dumbledore foretold the birth of the person who would be able to defeat Voldemort.

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... the one with the power to vanquish the Dark Lord will be born as the seventh month dies...

The prophecy was made in the Hog's Head when Trelawney was being interviewed by Dumbledore for the post of Divination teacher. The first part was overheard by Severus Snape, who reported what he had heard to Voldemort. She had no memory afterwards of making the prediction, and so never knew how important she was (OP37, HBP20, HBP25).

It could have referred to either Harry Potter or Neville Longbottom. Both boys were born at the end of July 1980 (Harry on the 31st, Neville on the 30th) and both sets of parents had escaped Voldemort three times. Voldemort chose to believe that Harry was the one meant and targeted him to eliminate the threat; in doing so he unknowingly actually set the prophecy in motion by "marking him as his equal" (as he had not heard the second part, he did not know of this danger) (OP37, HBP23).

Dumbledore knew that both boys would be in danger once Voldemort heard the prophecy, so he told Lily and James (and presumably Frank and Alice) to go into hiding. Unfortunately as the Potters made their Secret Keeper Peter Pettigrew , who was the traitor in the Order, this didn't stop Voldemort from hunting them down, murdering James and Lily and attempting to murder Harry (PA10).

During Harry's fifth year, the prophecy was the biggest focus of both Lord Voldemort and the Order of the Phoenix. Voldemort was obsessed with discovering the end of the prophecy as he believed it would tell him how to destroy Harry (this was the "weapon" that Sirius referred to in OP5) (OP37). A large part of the Order's work was trying to prevent this: they set up round the clock guard duty on the entrance to the Department of Mysteries. It was during this work that Arthur Weasley was attacked by Nagini. Voldemort attempted to steal the prophecy by placing the Imperius Curse on Sturgis Podmore and then Broderick Bode (not knowing that prophecies are bewitched so they can only be touched by those they refer to), and by first sending Harry visions of the Department of Mysteries to lure him there, and then a false vision of Sirius being tortured. Though this last attempt succeeded, the prophecy was broken and Voldemort never heard the remainder of the prediction (and even if he had, it would not have told him what he wanted to know).

Whether the contents of the prophecy were somehow leaked or someone just made a lucky guess, after the Battle of the Department of Mysteries there were rumours abound that it named Harry as the "Chosen One": the one destined to defeat Voldemort (HBP3).

• A record of the prophecy was kept in the Hall of Prophecy in the Department of Mysteries; this record was the centre of the Battle of the Department of Mysteries in 1996, when it was broken.
• The prophecy resulted in Trelawney being offered the post of Divination teacher, partly because it showed that she was a true Seer, and partly because Dumbledore knew that Voldemort would target her if she was not protected. This was also the reason Dumbledore made sure that she remained at Hogwarts when Umbridge sacked her, and continued in her post after Umbridge left the school (HBP20).
• It's likely that Snape bears a particular grudge against Neville (as well as an even larger one against Harry) because he knows that if Voldemort had chosen to believe the prophecy referred to Neville, Lily would still be alive.

"...and the Dark Lord shall mark him as his equal..."

Voldemort "marked" Harry both physically, with his famous lightning scar, and metaphorically, simply by choosing to go after him and seeing him as a true threat (OP37, HBP23).

"...but he will have power the Dark Lord knows not..."

The power Harry possessed was love (OP37, HBP23): his ability to love, the fact that he was loved, and the love that was on his side in the form of Severus Snape.

"...and either must die at the hand of the other, for neither can live while the other survives..."

Neither could live while the other lived, but neither could die while the other lived either: Harry was unknowingly safeguarding a piece of Voldemort's soul, and Voldemort kept the protection of Lily's sacrifice alive in both him and Harry by taking his blood (DH35). In the end, Voldemort was killed by his own rebounding Killing Curse from the Elder Wand. This could be seen as dying at Harry's hand firstly because the curse bounced off Harry's spell, and secondly because he was the master of the Elder Wand at this point (DH36).