There are two major plotlines. The first is the action plotline, a conflict caused by Cypher’s betrayal, that involves rescuing Morpheus from the Matrix. The second is the introspective plotline, in which Neo considers what his true potential is. In classic narrative tradition, these two plotlines follow concurrent development through the three Act structure.
Act One. The characters are introduced, and the plot is set up.
There are only five characters you have to remember, plus one that will be introduced in Act Two. The antagonists are indistinguishable guys in black suits called Agents. The protagonist is a computer geek named Neo. Also introduced are the mentor Morpheus, the love interest Trinity, the betrayer Cypher. Incidentally, there are other good guys, whom you don’t need to worry much about unless you’re really interested.
The other good guys are trying to rescue Neo, but we don’t really know from what yet. The Agents are aware of this, and they want to use Neo to get to Morpheus. They manage to capture Neo and plant a bug on him, but the good guys remove the bug and take Neo to Morpheus anyway. After making sure Neo wants to be rescued, Morpheus begins the process.
Act One Reversal. The world is not what it seems.
This is basically the big punchline of the film. The world as Neo knew it was a simulation, the Matrix, in which humans are enslaved, and from which he just woke up. The raison d’être of the good guys is to rescue other human beings from The Matrix. There is a base of operations called Zion, the last stronghold of humanity in the real world. We never see it, but we know it’s important.
Act Two. The rules are laid out and we find out more about Neo.
The good guys can sneak into and out of the Matrix to rescue other people, but while there, they’re vulnerable to the Agents, which are computer programs with great power inside the Matrix. The important point for the scenes later is that in the Matrix, if your mind is quick enough, you can get super powers. The dojo scene is the demonstration of this. Morpheus believes that Neo is the One, an individual who takes these super powers to a new level. Nobody else is sure of this, even Neo himself.
We also learn that there are robots in the real world that can attack the ship that all the good guys are on.
We see Cypher meeting with an Agent, meaning that he’s going to betray the good guys. Specifically, he’s going to turn Morpheus over to the Agents.
Everybody enters the Matrix again to take Neo to see the Oracle, the sixth character that I warned you about. The Oracle confuses the idea of free will a bit, but you needn’t worry about that so much. The important thing is that although he doesn’t believe the Oracle is all-knowing, Neo leaves convinced that he is not the One. This does not seem to faze Morpheus, however.
While they’re trying to leave the Matrix to go back to the real world, Cypher’s betrayal reaches fruition, and the Agents show up to capture Morpheus. They do, and all the other good guys manange to get away. Cypher gets away first and returns to the real world, where he starts killing good guys. Just as he’s about to kill Neo, a miracle happens and he’s taken out himself. Trinity and Neo return to the real world.
Act Two Reversal. Neo must put himself on the line.
Morpheus has been captured by the Agents, who can use a form of interrogation on him to extract information they need to destroy Zion. They can kill Morpheus and keep Zion safe, but Neo talks them out of it and insists on returning to the Matrix for a rescue attempt. Trinity goes with them. Neo also says outright that he is not the One.
Act Three. The conflict is resolved.
Neo and Trinity have to break into the facility where they’re holding Morpheus. They manage to get in, hijack a helicopter, and use it to get Morpheus out. The three of them run away, but the Agents follow. Morpheus and Trinity are able to leave the Matrix, but an Agent stops Neo before he can leave. Meanwhile, robots are approaching the ship in the real world, and they have to get Neo out before they can stop the robots. Neo has a fight scene with the Agent, then he has a chase scene, and then he is shot and killed.
Climax. Neo begins to believe.
Trinity reveals that she is in love with Neo, and that she believes he is the One. Neo comes back to life, and it is clear that he also believes in himself. He uses his super powers to their extreme and he destroys an Agent. The other Agents run away, Neo leaves the Matrix, and they stop the robots.
Denouement. Neo as savior.
Neo has wicked creepy powers now. He fully believes in himself, and he is confident in his ability to rescue people from the Matrix and win the war.