If real is what you can feel, smell, taste and see, then 'real' is simply electrical signals interpreted by your brain

Morpheus

Neo: I thought it wasn't real
Morpheus: Your mind makes it real
Neo: If you're killed in the matrix, you die here?
Morpheus: The body cannot live without the mind

Unfortunately, no one can be told what the Matrix is. You have to see it for yourself.

Morpheus

Throughout human history, we have been dependent on machines to survive. Fate, it seems, is not without a sense of irony.

Morpheus

[Neo tries a drink made by Dozer and winces]
Cypher: Good shit, huh? It's good for two things: degreasing engines and killing brain cells.

Neo, sooner or later you're going to realize just as I did that there's a difference between knowing the path and walking the path.

Morpheus

Have you ever had a dream, Neo, that you were so sure was real? What if you were unable to wake from that dream? How would you know the difference between the dream world and the real world?

Morpheus

Welcome to the real world.

Morpheus

Cypher: All I do is what he tells me to do. If I had to choose between that and the Matrix, I'd choose the Matrix.
Trinity: The Matrix isn't real.
Cypher: I disagree, Trinity. I think that the Matrix can be more real than this world. All I do is pull a plug here, but there... you have to watch Apoc die.

Neo: What did she tell you?
Morpheus: That I would find the One.

Oracle: You're cuter than I thought. I can see why she likes you.
Neo: Who?
Oracle: Not too bright, though.

Agent Smith: You hear that Mr. Anderson?... That is the sound of inevitability... It is the sound of your death... Goodbye, Mr. Anderson...
Neo: My name... is Neo.

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The Matrix Quotes

I'd like to share a revelation that I've had during my time here. It came to me when I tried to classify your species and I realized that you're not actually mammals. Every mammal on this planet instinctively develops a natural equilibrium with the surrounding environment but you humans do not. You move to an area and you multiply and multiply until every natural resource is consumed and the only way you can survive is to spread to another area. There is another organism on this planet that follows the same pattern. Do you know what it is? A virus. Human beings are a disease, a cancer of this planet. You're a plague and we are the cure.

Agent Smith

Neo: What are you trying to tell me? That I can dodge bullets?
Morpheus: No, Neo. I'm trying to tell you that when you're ready, you won't have to.