The phrase "The Unbearable Lightness of Being" is Kundera's own, but to understand it we actually have too start with Friedrich Nietzsche and the idea of "eternal return." Eternal return is the idea that our universe and our existence has occurred an infinite number of times in the past, and will continue to occur ad infinitum. In this theory, time is cyclical rather than linear. The idea of eternal return is an ancient one, but Nietzsche, a 19th century German philosopher, popularized it for modern times. That's why the narrator of Unbearable Lightness refers to it as Nietzsche's concept.
Nietzsche explored what the consequences of such eternal return would be. In his eyes, eternal return was das schwerste Gewicht, or "the heaviest weight." It was a petrifying concept to imagine that our lives have been and will continue to be repeated endlessly. But one could learn, through philosophy, to love the idea. The proper mind can embrace this weight, rather than be terrified by it. Nietzsche seems to conclude in Thus Spoke Zarathustra that we must live and act as though our lives functioned in eternal return, suggesting that we give our own lives meaning and weight by behaving this way. This brings in the concept of amor fati, or the love of one's fate. To embrace eternal return is, roughly speaking, to love one's fate. We talk more about fate in "Symbolism, Imagery, Allegory," so keep this in mind.
In Kundera's world, eternal return is a false premise. "Human time does not turn in a circle," he argues; "it runs ahead in a straight line" (7.4.13). So what are the consequences? Well, Nietzsche said that eternal return gives our lives with a sense of weight. So if our lives only occur once, it must mean that they are filled with lightness. This is where Kundera's phrase einmal ist keinmal comes into the picture. He translates this for us: "What happens but once, might as well not have happened at all. If we have only one life to live, we might as well not have lived at all" (1.3.18).
If we live only once, then we can never compare the decisions we make to any alternatives. And if we can never compare different outcomes, we can never know if the decisions we made are correct or not, which means we can never judge them properly or take responsibility for them. Hence, to live only once is to live with lightness. It's important to keep in mind that Kundera doesn't pose the question of whether life is heavy or light – he suggests that it is light, and then asks questions from there.
And the major question is this: which is better? Do we want lightness, or do we want weight? Which do we choose? Kundera takes a look at Parmenides, a Greek philosopher in the 5th century B.C. who considered the same question. Parmenides argued that lightness was positive and to be desired, while weight was negative. But the narrator of The Unbearable Lightness of Being isn't so sure about this. "The heaviest of burdens is […] simultaneously an image of life's most intense fulfillment," he says (1.2.4). "The heavier the burden, […] the more real and truthful [our lives] become" (1.2.4).
During the course of the novel, the narrator refers to the lightness of being in two different ways: the sweet lightness of being, and the unbearable lightness of being. A few characters are able, momentarily, to revel in the sweet lightness of being. A key example is Tomas, after Tereza leaves him alone in Zurich and returns to Prague: "Suddenly his step was much lighter. He soared. He had entered Parmenides' magic field: he was enjoying the sweet lightness of being" (1.14.7). For two days, he feels the "sweet lightness of being rise up to him out of the depths of the future" (1.15.4). For it only lasts for two days before he is "hit by a weight the likes of which he had never known" (1.15.4), namely, his compassion for Tereza.
The narrator first uses the phrase "unbearable lightness of being" when he refers to Sabina, just after she's left Franz in Geneva. "And Sabina–" asks the narrator, "what had come over her? Nothing. She had left a man because she felt like leaving him. Had he persecuted her? Had he tried to take revenge on her? No. Her drama was a drama not of heaviness but of lightness. What fell to her lot was not the burden but the unbearable lightness of being" (3.10.2).
Kundera argues that lightness is unbearable, but it is up to us as readers to understand the reasons behind this argument. What makes lightness unbearable? Remember Nietzsche's interpretation of eternal return? It's scary, almost paralyzing, to think about eternal return. But on the other hand, it means our lives have meaning, significance, weight. And we can learn to love that. Conversely, lightness may seem at first to be a sweet deal – no responsibility, no judgment, no meaning. Sounds like fun – at first. But eventually, we would like for our lives to mean something. We want them to have weight and significance, because we want them to matter. A great example of this is Tomas's two-year holiday from medicine. At first, he revels in the freedom of having no responsibilities. But after enough time has passed, he decides that two years is about as long as he can stand being on holiday. We're guessing you'd feel the same way after vegging out on your couch for a month.
The problem is, try as we might to give our lives weight…we cannot. Our lives are fundamentally light precisely because they occur only once. This is evidenced by the four deaths of the four main characters, which we talk about in "What's Up With the Ending?"
Kundera's argument, then, is two-fold.