This book was published under the pseudonym "Anti-Climacus," and that narrator introduces the philosophy of Christian Existentialism by mentioning a story where Jesus Christ tells his friends and family that Lazarus's illness is not "unto death," but, Lazarus does die. This sets the stage for the greatest miracle of Jesus's ministry: Raising Lazarus from the dead.
Anti-Climacus describes this things and then starts his philosophical questioning. If Jesus had not risen Lazarus from the dead, would his claim, "This sickness is not unto death," have been true or false? The question is difficult because it brings up an interesting point about death—because no one knows what death brings, there is no way to know whether death on this planet constitutes actual death, or whether there is a translation of sorts into a new kind of reality. Ultimately, this discussion leads Anti-Climacus to the conclusion that the true "sickness unto death," is despair.
Anti-Climacus describes this despair as a kind of misalignment. When living life, we realize that there are many commonly held assumptions about life that technically are unfounded, such as the belief that we need to earn a lot of money, or become famous or something. Anti-Climacus says that this is essentially a dialectical problem, because the reality of life needs to be synthesized with a hypothetical, ideal understanding of reality.
Then the book offers three analyses of this despair. First there is the inauthenticity despair, caused by cognitive dissonance. This despair is the subtle terror of knowing that no matter what we do, we're imposters playing out certain roles. In other words, the first terror is the despair caused by not knowing one's true identity or destiny.
Then there is the despair caused by limited self-acceptance. Anti-Climacus describes this second despair as a kind of false assumption, where a person believes that their experience of identity (as one human) is their entire identity. He argues that we should consider our "self" as a broader idea that extends out into the reality of our lives. Our "self" is not only the primate body we live in; it is also the entirety of our human experience of reality (says Anti-Climacus).
Thirdly, and finally, there is the despair caused by the infinite nature of life, compared to the finite nature of the human experience. Ultimately, this has many facets, but Anti-Climacus boils it all down to one single point: The third despair is essentially the terror caused when a person cannot admit to themselves that they are utterly dependent on others for love and acceptance. Simply put, the final despair is loneliness.
