A response essay, believe it or not, is exactly what it sounds like! You are expected to write an essay in which you discuss your response to a literary text you have read. The text might be an essay, an article, a poem, a short story, or a novel. The idea is to focus on what the text made you think and feel.
The structure of such an essay is not going to be all that different from the structure of any other essay. You need to write an introduction, providing the name and author of the text you are responding to and a brief summary of that text. From there, you can write a thesis statement that explains the points you will develop in your body paragraphs. Each body paragraph should focus on just one point. And finally, you will write a conclusion that summarizes your points.
Now, what points might you cover in a response essay? You might discuss whether or not you agree with an author and why. You might discuss some favorite passages from the text. You could discuss how the text influenced your thinking and your feeling on a particular matter. You could also discuss the text's connection with some other reading you have done. Or you could provide an example of a way that the text connects with your own life, personally, academically, or professionally.
A response essay is important because it is a way of making meaning of what you read. If you have no response at all, then you are not making meaning of the text, and that would be a shame because that is the point of our reading anything!