In John 3, in part of Jesus’ extended colloquy with Nicodemus, he says that believing in him—believing that Jesus is who he says he is—makes zōen aiōnion available. What does that mean?
Literally, it means “the life of the ages.” There are several ways of understanding that. You can think of the reference to “ages” as meaning a long period of time. Or you can think of it as a higher plane of life. “The life of the ages,” in this way of construing it, is true life. It is not the succession of meaningless ephemera that pass for “life” for those who know only what they see. It is the higher plane where what is True in always being enacted and always in danger. And Jesus represents that “higher plane.”
These two meanings offer us zōen aiōnion as an infinite extension of the life we are living now—a life that lasts forever—or, alternatively, as access to another kind of life entirely. To me, the second meaning–another kind of life entirely–makes a great deal more sense and it solidifies the case, I think, that Nicodemus can find no way to grasp that notion at all.
That’s as far as I can take the conversation by relying on how the words might be understood. I would like to offer a more visual explanation, however. It is the world presupposed in The Matrix, a 1999 movie by the Wachowski brothers. The only world we see for a long time in the film is entirely illusory. As viewers, we don’t know that, but we get a clue when the bad guys ram a truck into a telephone booth where Trinity (!) is making a call and, not finding her body there afterward, they say, “She got out.”
“Out?” Hmm. She got “out” of “here,” presumably by going “there.” Where is “there?”
We find out where “there” is when Neo (!) is unplugged from the vat of goo where his body has spent its entire life so far and is flushed down into the sewer. The people in his apparent life have known him as Mr. Anderson, but he has taken Neo as his nom de guerre. When his body hits the water in the sewer system, a cable with a grappling claw is lowered by a hovercraft canned Nebuchadnezzar IV. It clamps onto this body and reels it up into the plane. That is where “there” is. It is where Trinity went when she escaped the phone booth and she is there waiting for him.
“Out” is where the life of the body and the life of the mind are one thing, rather than two. That is how we learn what the Matrix is. We do know what the Matrix is, in a sense, because Morpheus, the captain of the Nebuchadnezzar IV tells Neo what the Matrix is. It is, according to Morpheus, “the world that has been pulled over your eyes to blind you to the truth.” But when you are in the hovercraft, you are out of the Matrix and you are not blind anymore. 
Being “out of the Matrix” means, for one thing, that the sensations you have— what you feel and see and smell—are not provided by a brain probe the way they were when your body was “living” in the tub of goo. The sensations you now have are provided by your fingers and your eyes and your nose. So the life of your mind and the life of your body are intricately connected, not merely simulated. You are living the life you think you are living.
But there is another thing it means. It means that you can be in actual, not just imagined, mortal conflict with the Matrix as a system. It means that your life can mean something. It can be the result of the choices you make. It can be “your life,” not just in the sense that you intend it, but also in the sense that you can live out the life you have chosen. 
Of course, the life you have chosen was not available to you before. Someone has to come from “out there” to contact you. It was Trinity in the case of Neo. She did things he knows cannot be done (she took over his computer and used it to talk to him) and then, after arranging a meeting with him (in the the Matrix, of course, the only “place” available to Neo) tells him that there is an answer to the most important question he has ever asked, the question that has dominated his life up to this point: “What is the Matrix?” And Morpheus symbolized the choice for Neo as blatantly as Jesus did for Nicodemus. The red pill or the blue pill? Continued blindness or coming to grips with the truth. Choose.
So Neo, after his body is reeled up into the Nebuchadnezzar IV, begins to experience “the life that means something,” arguably, the zōen aiōnion. It is awkward at first. Lying there on the table, too weak to get up, he asks Neo, “Why do my eyes hurt?” Morpheus replies, “Because you have never used them before.” And immediately, we understand how that is true. All the optical stimulus Neo has ever received has not come from his optic nerve, but from the brain probe that simulated “seeing something.” He has never before seen what was actually there because nothing was actually there in the “life” he seemed to be living.
In the “life of the ages,” the zōen aiōnion, something is really there and you can see it with your own eyes. And, just as your vision is now authentic, so now your intentions can be authentic, You can life the life you choose—remembering always that it chose you before you had any possibility of choosing it—and all of your choices presuppose a war against the Matrix. The little band of people who rescued you from the Matrix is a band of soldiers at war with the Matrix and now, you too are at war with the Matrix.
The dominant characteristic of the life of the ages is that you can make real choices. The Matrix is constitutionally hostile to such a life, so you cannot choose peace with the Matrix. You can fight it, as Neo does, or you can surrender to it, as Cypher, one of the little band that rescued Neo, does. Cypher asks to be “re-inserted” into the Matrix, knowing it will be the last authentic choice he ever makes.
The film, The Matrix, and its two lesser sequels, presuppose the disjunction between what we think of as “life” and “real life”—a life that matters. I think that in John’s gospel, the disjunction between the kind of life Jesus offers Nicodemus and the kind he is “living” at the time he goes to talk to Jesus, is a choice as stark as that. The Life of the Ages is clearly not, in The Matrix, continuing to live for a very long time, the kind of life you think you are living. It has to mean living a life on a different plane entirely;  it has to mean a life that means something, however long it lasts.
I think that is the reality that The Matrix points to. It is the most visually persuasive representation I have ever seen of the disjunction between what Nicodemus was talking about and what Jesus was talking about.
 This is the most vivid representation I know of the distinction Paul makes in Romans 12 when says we are not to be “conformed” to the world, but to be “transformed.” Neo, in his imaginary life, his life within the Matrix, was a hacker and he was very anti-Matrix.” But as long as his body and therefore his mind was actually in the Matrix—imagining the life the brain probe constructed for him—there was no possibility at all of being “transformed.” That happened when he was rescued from the Matrix by the cable and the grappling claw.
 Neither of these “lives,” by the way, lasts forever. When your body is “in the Matrix,” you are valued for the voltage your body produces and which the Matrix uses as a power source. When you are outside the Matrix, you might be killed because you are after all, an enemy of the System. But even if you are not killed, your body will still die in the normal way. Living a real life doesn’t give your cells extra telomeres. But when you die outside the Matrix, you die having actually lived.
 No pun intended, but it is hard to avoid it.