What is the Analytic of the Sublime by Kant?

This refers to the second book of the first division of Kant's Critique of Judgement. Here, Kant discusses aesthetic judgement and breaks this down into two components: the beautiful and the sublime. Beauty, for Kant, is a pleasurable feeling caused by the perception of an object that everyone ought to recognize. The sublime, on the other hand, is a feeling of pleasure in the superiority of reason over nature, on the one hand, and the displeasure of the imaginative or physical powerlessness in the face of nature. In this way, appreciation of the sublime is a kind of "vibration" between exaltation and dread. It is this quality of displeasure that marks the difference between the sublime and the merely beautiful.

The Analytic of the Sublime is a concept introduced by Immanuel Kant in Critique of Judgment to delineate the nature of beauty as opposed to the nature of the sublime. While they are similar, in the study of Aesthetics, Kant reasons that Beautiful and Sublime are very different concepts.
Kant's explanation of Sublime vs. Beautiful hinges on the lynch pin of human reasoning, making Sublimity something that is beyond the understanding of lesser beings. Beauty is intrinsic, he reasons, and all creatures can find beauty, to an extent. Sublime, however, is something that passes beyond the realm of our understanding—it is not only beautiful, but powerful and mysterious in its incomprehensibility. Because of this, humans alone can experience it because we are the only creatures that have a rationality that knows its own limits. In this way, sublimity is something that is so beautiful and powerful that it breaks through the limits of our understanding and goes to the core of what makes us human—reason.

The Analytic of the Sublime comes from Book Two of Kant's Critique of Judgment. This is a very important work in the field of aesthetics, a branch of philosophy concerned with matters of beauty and artistic taste. In the Analytic of the Sublime, Kant attempts to draw a clear distinction between the beautiful and the sublime. Most people tend to confuse the two, but Kant wants to show that the popular understanding of the sublime is mistaken. Kant's argument is an analytic in that it breaks down the component parts of what constitutes the sublime the better to analyze them.
So what does Kant mean by the sublime? The sublime, unlike the beautiful, has an air of mystery about it, something we can't quite seem to grasp or define. Things that are sublime, like a large black storm cloud or a gigantic mountain, appear too big for us to comprehend. However, on closer inspection, this turns out not to be the case. Our whole concept of sublimity is related to our understanding of absolute freedom—for example, the moral values that, according to Kant, we give to ourselves out of our reasoning faculty—and absolute totality (in simple language, the whole of everything that exists).
The sublime, therefore, is related to our faculty of reason, something unique to humans. As such, it is in us, in our minds' capacity to apprehend truths of reason, rather than in the objects themselves, such as the examples of the storm cloud and the large mountain we looked at earlier. Whenever we feel ourselves overawed by the sublime—being caught in the middle of a violent sea storm, for example—what's actually happening, according to Kant, is that we're getting in touch with those ideas of reason, those ideas of absolute freedom and absolute totality which we as rational human beings all share.