Continuity mistake: After the helicopter smashes into the building and Neo saves Trinity, they look at each other and you can see smoke from the building. After that, Morpheus is talking to Neo. You can see several shots of him and Neo talking and behind them, the building and its windows are still intact.
Continuity mistake: During the rooftop scene check out the tiling situation. When Neo takes two bullets, he falls to the ground. He lies mainly inside one tile, even the left gun fits in. (And it's on a vertical line) His top hand lies just on the line. Now, watch again when the agent is dead. All of the sudden he's in-between two tiles, and the gun is now on a horizontal line, but not a vertical.
Continuity mistake: When Agent Smith is seen the first time, he is talking to the old cop outside, then begins to walk away and the other agent follows. The cop turns his head to continue to look at the other agent when he is walking away or past him, yet in the following shot the cop has his head turned, but the other agent is still in sight and is just barely walking past him.
Continuity mistake: When Neo is in the helicopter firing the automatic weapon into the building he is moving the gun horizontally. The bullets strike vertically, though, starting from the floor near the window and traveling upwards towards the back wall of the room. There are two shots that show this.
Continuity mistake: Near the start, when Agent Smith jumps over the building and the police officer in the front says, "It's impossible," look closely in the background when the agent lands. You can see it's a different officer than the one that said it.
Visible crew/equipment: At the beginning of the Subway fight there's a shot of Morpheus, then of Neo, then in the next shot of Agent Smith the white screen is visible in the right lens of his glasses.
Other mistake: When Neo shoots with the chopper's machine gun the window smashes from a single point in the middle, shattering two panes. Water then starts splashing up from bullet impacts right at the base of the window, but the glass doesn't smash anywhere around there - it could only work like that if Neo was shooting down at an incredibly sharp angle, which he obviously isn't.
Continuity mistake: Near the middle of the film when we see Morpheus and Neo have entered the Matrix in a scene that begins at a city intersection with a "Walk/Don't Walk" Traffic Sign. Just as the cross the street, Morpheus is speaking to Neo, the next shot looks over Neo's shoulder back towards the intersection, it is no longer visible, as though they have traveled farther down the block than physically possible in the time allotted.
Continuity mistake: During the fight scene in the train station between Agent Smith and Neo, after a lot of fighting Neo is thrown back and throws up blood. He then gets up and wipes it off with the back of his hand. Straight after that, it isn't visible anymore. It's completely vanished.
Continuity mistake: At the beginning of the film, when Neo is having the conversation with his boss about his arrival time at the office, when the camera is on Neo, his hands are behind his back, when the is on his boss (showing Neo's back), Neo's hands are in front of him. They're all instant cuts - no time for him to move.
Visible crew/equipment: Bad idea to have mirrored sunglasses in the movie. Look into the lenses of the agents, Neo, Trinity, and any one else who has them, and you can see the camera perfectly.
Continuity mistake: In the lobby scene, after the SWAT team has entered, one of the SWAT guys charges, firing his M16 at Trinity. After she kicks him while he's putting in a new clip, his right hand's empty in the close up, but is holding the M16 (which he then drops) in the wider shot.
Factual error: When Agent Brown shoots at Neo on the skyscraper, the last shot scrapes Neo in the knee and on his shoulder. This would cause both shots to be vertical when standing up. However, when Neo stands up and indicates to the scrape, the shoulder scrape is horizontal - not possible given the trajectory Agent Brown is firing from.
Continuity mistake: In the fight in the toilet between Morpheus and Agent Smith, they're on the floor and Smith has his glasses on. Smith prises Morpheus' hands off him, punches him back against the wall, stands up in a menacing way, and his glasses aren't on any more. He wasn't hit at any point in that brief exchange, so where have his sunglasses disappeared to? [Only in the UK version - the BBFC's got a thing against headbutts, so cut several out from the film. In this case it caused Smith's sunglasses to disappear for no reason.]
Other mistake: They are presumably in the US - trying to make it look that way, anyway. Cars driving on the right, speaking English etc. However the filming location was overseas, revealed by the elevator in the lobby scene. When they hit the button, the plate refers to the elevator as a lift.
Character mistake: When Morpheus is explaining to Neo the reason the Matrix exists, he states that a human being gives off 25,000 BTUs of body heat to help power it - this is just not true. Even human beings that are running a marathon can't give off that much heat, and the humans in the Matrix are all lying down, not moving.