Thomas Anderson, computer programmer and hacker, ponders the cryptic question: "What is the Matrix?" He is soon contacted by Morpheus, just in time to not quite avoid capture by three Agents. The Agents, fitting the 'Men in Black' look perfectly, detain Mr. Anderson (better known as Neo to everyone but agent Smith) and ask him to help them find Morpheus. When he refuses, the agents melt his mouth off and let a bug the size of a Wiimote crawl inside his belly button.
Shortly thereafter Morpheus contacts Neo again, this time to meet in person. He offers him a choice: take the blue pill, and just deal with the trauma of the crazy shit that's already happened, or take the red pill and see if shit can get any crazier. Neo, totally unfazed by the notion of accepting pills offered to him by strangers, takes the red pill. He wakes up outside the Matrix, where Morpheus shows him the truth: reality as we know it is a simulation beamed into our brains.
While the machine uprising was pretty much to be expected (see 5 Terrible Life Lessons Hollywood Loves To Teach You), the reasoning was unique: after we blackened the sky to take out the machines' solar power, operating under the pretense that agriculture is for pussies, they responded by trapping humans in pods and using them as power (because humans at rest give off way more energy than they take in.)
Pictured: A reliable energy source.
After having martial arts uploaded into his brain, Neo quickly learned that in the Matrix, you can bend the rules of the virtual world, granting you superhuman powers. Neo, however, is even more superhuman than the rest of them; Morpheus tells him that he is The One, a person who can change literally anything inside the Matrix. As such, Neo is the only one who can kill an Agent, the reigning badasses of the Matrix until Neo showed up.
Morpheus then took Neo to see the Oracle, a clairvoyant program with a penchant for baking cookies, to determine his fate. She told him that he wasn't ready to start his being The One yet. On the way home, Morpheus was captured by Agents, and Cipher, a member of the ship's crew who felt pod life was the good life, killed most of the crew, leaving only Morpheus, Neo, and Trinity alive.
Neo And Trinity started a rescue mission to save Morpheus when an Agent found them, and tried to shoot Neo. In one of the most famously awesome movie moments of all time, Neo dodged the bullets with a wicked backward lean, terming any cool dodge henceforth as 'Matrixing that shit'. Neo and Trinity then broke into the building where Morpheus was being held, and began a maelstrom of slaughter that cemented Neo's status as one of the 6 Great Action Heroes (Who Should be Convicted of Murder).
He and Trinity then boarded a helicopter, shot the hell out of a few Agents with a minigun (one of 7 Items You Won't Believe Are Actually Legal) and got Morpheus out of there. Neo fought, and defeated, Agent Smith. Being an Agent, Smith simply stole a new body and Neo decided getting the hell out of there was a better plan.
Not that it did him any good. After a long chase, Smith (in yet another new body) unloaded a clip full of Desert Eagle rounds into Neo's chest. This killed him, but Trinity, who had fallen in love with Neo (after being told by the oracle she would fall in love with The One) kissed him. This prompted a respawn for Neo, who promptly made three Agents his bitch with minimal effort.
The second movie in the franchise opens with Neo's dream of Trinity getting killed by an agent. Right after he wakes up he goes to a meeting of ship's captains discussing the machine attack on Zion, who collectively decide Zion is jolly well fucked if the scans are right and there are actually a quarter million Sentinels on their way, even though Sentinels are one of 8 Classic Movie Robots That Actually Suck At Their Job.
Due to this news, Commander Locke, the HNIC of the Zion army, ordered all ships to return to Zion though Morpheus had a ship stay behind to await contact from the Oracle, because who are you gonna trust? The greatest military mind humankind currently has at their disposal, or some lady with candy who admits being a part of the other team?
Since his ship was running on fumes, Morpheus couldn't actually stay behind and he came home, and Locke was naturally upset by his defiance. As punishment, he made Morpheus break the news about imminent death to the people of Zion. He was met with resounding cheers, and the city actually threw a party. Locke wept the bitterest tears, but then banged Morpheus' ex-girlfriend and felt infinitely better about himself.
"Suck on THAT, Morpheus."
Neo received word from the Oracle. They go into the Matrix to see her. She tells him he needs to return to the Source (the Matrix server, if you will) but he'll need to get the Keymaker first from the Merovingian, a program that is basically every American stereotype about French people personified. As Neo tries to leave, he runs into Smith, who is no longer Agent Smith. He, too, has woken up to the fact that it's all a computer program, and the first thing he did with that knowledge was Ctrl-C.
And then he spammed Ctrl-V like he was getting paid to do it.
One Neo realized there were literally hundreds of this dude to fight against, he ran away again. Except this time, he flew away (which admittedly makes it a lot less lame).
The group goes to the Merovingian, who tells them to fuck off as politely as possible. The group sulks away, when the elevator stops unexpectedly. The Merovingian's wife, Persephone winds up helping the group find the Keymaker in retaliation to the Merovingian cheating on her.
But you know what? If you're cheating on this, maybe you deserve to get your Keymaker stolen.
The Merovingian promptly sent a shitload of people to take back the Keymaker. While the Twins, a pair of guys in white suits who could phase, chased Morpheus and Trinity who had the Keymaker, Neo kicked the shit out of every single other guard they had. While Neo continued to cut a bloody swathe through the Matrix, Trinity and Morpheus, accompanied by the Keymaker, were being chased in what is one of the most awesome, if not the most awesome, car chase scenes in the history of cinema.
After a flying Neo saves Morpheus and the Keymaker from a giant explosion, the crew of the Nebuchadnezzar, along with a few other ships, prepare to help Neo get to the Source. They plan to do so by destroying two power plants, to avoid Neo's opening the door setting off an earth-shattering kaboom. Neo meets the Architect, who delivers an extraordinarily long and complicated speech. In layman's terms, he tells Neo that his powers are the result of an imperfection in the coding of the Matrix. He further explains that the machines are going to wipe out Zion - for the sixth time, no less - and the only way to stop humanity from dying altogether is to return the hax which he carries back to the source and pick a group of people to build Zion for the seventh time, just so they can eventually get killed, and the next One can pick out a group of people, and so forth.
The Path of The One.
Neo gave him the finger, went back to the Matrix, and saved Trinity from the death he envisioned in his dream. Upon saving Trinity, they emerged from the Matrix just in time to flee their ship, due to the Sentinel attacks. The Sentinels found them, and Neo stopped them with his hand, showing that if you get awesome enough at a computer simulation, you can actually take the stuff you used inside of it into the real world when you're done.
What a truly terrifying notion.
The third and final installment of the matrix trilogy opens with Neo laid the hell out, because even though his Matrix skills now have a dash of IRL, they're apparently quite hard on him. His body is next to Bane, a member of a ship's crew who fell victim to Smith's copying, allowing the Agent to enter the real world. Neo's mind, however, is trapped between the Source and the Matrix. Morpheus and Trinity learn the Merovingian is the only one who can get Neo out, and head to his club to make damn sure he does. When they arrive the Merovingian says he wants the eyes of the Oracle in exchange for Neo, and Trinity counter-offers with not shooting him in the face in exchange for Neo.
Neo speaks to the Oracle after getting out of the train station, who explains that he has a connection to the Source, which 'explains' his ability to stop a bunch of robots the size of a car with a thought. She also explains the duality between Neo and Smith, and that before long one of the two is gonna be running things. Not long after Neo leaves, Smith enters the Oracle's room with five of himself as bodyguards. Smith lets her know that resistance is futile and she will be assimilated.
And he was right.
Meanwhile, back in the real world, Neo and Trinity take one hovercraft to the Machine City while the cews of both hovercrafts take the other one. Niobe, who apparently rules at driving these hovercrafts, planned to drive the Hammer down mechanical pipelines - which is about equivalent to driving a semi truck at 60 mph down a series of alleyways - while being chased by a group of Sentinels.
Back in Zion, the army of Sentinels breached the city walls and promptly began wrecking the shit out of everyone and everything. A kid - literally, his name is Kid - hopped in a mecha and shot up a chain, letting the Hammer crash through, set off an EMP, and wipe out all of those Sentinels, promptly saving the day.
Except for the fact that the EMP actually fucked up everything in Zion, as well. And a bunch more Sentinels were on their way.
Fortunately, Neo got to the Machine City after a nasty crash which killed Trinity and trashed the ship. Okay, maybe not fortunate for Neo, but fortunate as all get-out for the people of Zion, because he managed to stop the Sentinels from killing all the people while he fought Smith for control of the Matrix. In exchange for peace, Neo offered to get in there, kill off Smith, and let the machines have their Matrix back.
After allowing Smith to assimilate him, then promptly making him and all of his copies explode, the Matrix reset from a rainy wasteland to what it was before Smith kinda ruined everything. And because Neo had kept his end of the bargain, and the machines couldn't go back on their word, he attack on Zion was called off and the humans were allowed to live their lives in peace. Well, the ones who weren't already killed.
On top of the movies, the Matrix gave rise to three video games: 'The Matrix Online', a Matrix-based MMORPG, 'Enter the Matrix', a game with a severe lack of getting to play as Neo, and 'The Matrix: Path of Neo', the game to remedy that problem. A collection of anime and anime-like videos based around other in the Matrix, called 'The Animatrix', were also released.