Been a long time since I read but IIRC:

1. The final reveal from the Pensieve and Snape's memories is that there are actually eight Horcruxes. When Voldemort tried to kill Harry as a baby, the murders of the Potter parents and love protection that caused the Killing Curse to backfire split Voldemort's soul again, and it latched onto infant Harry. Harry is himself a Horcrux. So Voldemort can't die until Harry himself is dead. So he goes to face Voldemort in the Forbidden Forest and is prepared to sacrifice himself. Voldemort "kills" him.
2. Dumbledore actually lied again. Harry and Voldemort basically get stuck in limbo, where he has some kind of magical conversation with Dumbledore (I'm not sure this is ever explained). Voldemort's Killing Curse once again failed to kill Harry, and instead destroyed the portion of Voldemort's soul latched onto Harry (again I don't think a Killing Curse was originally able to destroy a Horcrux and this is also never really explained). Basically, just as Voldemort was unable to die while Harry lived, so too Harry cannot die while Voldemort lived. But now the last ties between them have been destroyed.
3. Thing is, there's a problem: one more Horcrux still exists, the snake Nagini. Both Harry and Voldemort wake up. Harry continues to fake being dead. Voldemort parades Harry's "corpse" before the remaining defenders of Hogwarts to try and break their morale once and for all. Neville defies him, so Voldemort tries to torture him by making him wear the Sorting Hat and setting it on fire. While everyone is distracted Harry disappears, and once this is noticed all hell breaks loose. Neville suddenly draws the Sword of Godric Gryffindor from the burning Sorting Hat and fucking kills the snake, because he's the real hero of the entire series.
4. Battle is joined (EDIT: I actually totally forgot about the "love protection" aspect of Harry's earlier sacrifice - because Harry sacrificed himself under the impression he was going to die, his non-death still conveys love protection to literally everyone on his side in the battle and they wipe the floor with the remaining Death Eaters) and Harry and Voldemort confront each other one last time in the Great Hall. Voldemort, all arrogance, intends to kill Harry once and for all. Harry tries to I guess "redeem" Voldemort one last time by warning him that the situation is not what he thinks it is and if he tries to kill Harry, the curse will backfire. Why is this?
5. The language governing ownership of the Elder Wand is deliberately vague. The parable implies that you have to kill the previous owner of the Wand. We (and Voldemort) have thought the chain of ownership of the Wand went Dumbledore-Snape-Voldemort. Snape killed Dumbledore at the end of the last book. Voldemort killed Snape earlier and this allowed Harry to get Snape's memories, etc. But the truth is you only have to "defeat" the previous owner. This fundamentally changes the chain of ownership. At the end of book 6, when Harry and Dumbledore reappeared on the tower, the latter was confronted by Malfoy. Malfoy used the Disarming Spell on Dumbledore. In magical terms, this means Malfoy "defeated" Dumbledore. This actually screwed up an additional gambit by Dumbledore to destroy the power of the wand - he was already dying from the ring-Horcrux's curse, so he colluded with Snape to allow himself to be killed by the latter; if Dumbledore was "defeated" by his own design, he theorized it would break the cycle and destroy the power of the Wand for good. But Malfoy's actions threw a wrench into everything. Malfoy couldn't bring himself to kill Dumbledore, so Snape stepped in as planned. This fooled Voldemort on so many levels.
6. Halfway through book 7, Harry and co are captured by the Malfoys. In their escape, Harry used the Disarming Spell on Malfoy, "defeating" him. This means the actual chain of ownership for the wand is Dumbledore-Malfoy-Harry. Harry is the current master of the Elder Wand. If Voldemort tries to kill Harry with the Wand, the Wand will refuse to kill its master and backfire on Voldemort. Voldemort, all arrogance, refuses to believe this. He tries to kill Harry, the curse backfires yet again, and it kills Voldemort instead.
7. Harry takes the Elder Wand, uses it to repair his old wand, and then basically sticks it in a secret vault in Dumbledore's office presumably to sit there forever. He hopes to end the cycle his own way by dying peacefully in his bed (this is incredibly at odds with him presumably pursuing a career as an Auror/Wizard Fed, but w/e)

This shit was gripping when you were younger but if you step back and view the series with a wider critical eye it has a lot of gaping problems. Rowling basically introduced several incredibly important McGuffins crucial for resolving the entire series in the last book and retroactively tied them in with earlier events and foreshadowing. Voldemort's final defeat was certainly thematically appropriate (he was so arrogant and unable to comprehend he was mortal he refused to listen to an honest attempt to warn him he was wrong) but Harry basically wins on a technicality. A better series, if it really wanted to end this way, would have spent a lot of time in the series setting up these McGuffins and the ambiguity of the language regarding ownership of the McGuffin. But it's thrown in at the last minute like an Ass Pull.