We can see in this video clip from The Matrix that, when chased by the agents, Neo confuses right and and left:
When Neo asks the operator Tank to help him find a way to escape, Tank tells him to take the door on the left, clearly saying "on your left". Neo tries to open the door on the right, which prompts Tank to tell him "Your other left". Using that door, Neo manages to escape from the agents of the Matrix.
This scene always puzzled me. This scene does not seem to have any use and we know that the movie had a tight budget (using decors from Dark City). So any unnecessary scene was probably removed from the script. In addition, the Wachowskis are known to have written a scenario, where most, if not all, elements have a purpose and an intended symbol.
Out of universe, why is this scene in the movie?
Is it a blooper from Keanu Reaves that was not removed from the final cut (no time or money to reshoot the scene)? Or was it intended and means something about Neo? A way to increase tension during the chase scene?
Because it showcases that Tank sees Neo. He is not limited to what Neo tells him on the phone, he is actually tracking his position.
Until then, it would have been possible that the operator is not directly aware of where the others are, they could be basing themselves on what they've been told (1), or what they can "read from the Matrix" (2).
But Tank telling Neo that it's the "other left" means that Tank has more knowledge than what is being communicated over the phone.
Note that Tank had already previously been shown to be somewhat aware of where the agents are (as they are part of the Matrix itself), but not necessarily where his team is. That's the point of the scene, it disambiguates an otherwise ambiguous factoid about what the operators can and cannot see.
Secondly, it's also a short moment of comedic relief.
Thirdly (but this is really reaching, imo), it showcases that Neo is running for his life and not mentally calm, as he is making mistakes and acting instinctively.
This isn't a blooper. The scene exists in the 1997 version of the script (but not the 1996 version where Neo travels to the exit without Tank's assistance) as well as the final draft screenplay seen in The Art of the Matrix
TANK: Shit! The door on your left.
Neo lurches, kicking in an apartment door.
TANK: No! Other left!
He whirls back to his other left, battering through the DOOR which SPLINTERS, PERFORATED by BULLETS.
The final draft notes that the building is a maze with zigzagging corridors. It's easy to get turned around, especially when you're being shot at.
(I don't have a source, only my interpretation, and a hint from the 1996 script about the state of Neo's powers by the time of the chase. Given the compressed timeframe of the final script, I take this to be a conservative estimate of Neo's level of awareness by this point in the movie.)
All of Neo's powers flow from the same place: an ability, on some level, to perceive and interact with the raw, unfiltered code of the Matrix.
Neo's superhuman reflexes are not in fact superhumanly fast - they are a cheat. They come from sensing and responding directly to changes in the Matrix, without waiting for this information to be filtered through a process of slowly and sloppily encoding and interpreting a series of audio-visual cues. (For gamers: it's essentially the same as the Psycho Mantis trick from Metal Gear Solid.)
As Neo's powers grow throughout the movie, as his reactions get faster and faster, all that is really happening is that Neo is becoming more and more aware of the Matrix itself and less and less dependant on the audio-visual illusions it casts.
I believe that the "other left" scene is a nod to Neo's level of awareness by this point in the movie. It is contemporary to a scene in the 1996 script:
Looking over his shoulder, watching the mirrors, Neo drives almost unconsciously. (1996 script)
In the scene as originally scripted, Neo is driving to the exit by car, apparently without needing to look where he is going, because he has another way of sensing what is ahead of him: his awareness of the Matrix. Worried more about the pursuing agents, Neo's focus is behind him rather than in front.
My interpretation of the "other left" scene is that it turns this scenario around - instead of, in the original script, physically looking behind him and "sensing" in front, Neo is (more comfortably) physically looking ahead of him as he runs, but is "sensing" behind him, looking for the pursuing agents.
Because his focus is still on the agents, when he hears "left" he moves to the "left" that he perceives most strongly.
The top voted answer is great. Considering it to be a presentation of Tank's view, and abilities makes sense. Comic relief is also very true. But for me, this scene has always been about Neo's humanity as well. He is not a mathematically programmed machine, and this simple mistake is one that humans do quite often in the real world. Making him go the wrong way in a stressfull situation, simply makes him more relatable to me as a viewer. It helps separate him from the machines.
Don't know for sure and I have no source to back out up, but I'd say it's there to enforce supporting the fact that they can watch him from a third person perspective and he somehow assumes they see him from the front since he's running towards them.
Also "on your left" doesn't necessarily imply "from your perspective", so he doesn't really confuse it. The information is simply too vague given the context.
A bit of over analyzing going on here I think.
It's a moment of humor in an otherwise extremely tense situation... akin to Neo's line "Mr. Wizard, get the hell out of here!". An homage to the "Tooter Turtle" cartoon.