An "elastic" form of prose writing, according to Elizabeth Hardwick (American, b. 1916), essayist par excellence. The English word "essay" has its origin in the French word essai, which means to try, to attempt, and thus among the few agreed upon qualities of an essay is that it is a writer's attempt to explore an idea, a question, a dilemma, an issue, a phenomenon, etc. "Exploration" is another quality of an essay; the essayist most likely to hold a reader's attention is on a voyage of discovery, and surprising him or her own self in the process is often what makes an essay lively and compelling for the reader. The writer's presence in the essay distinguishes the form from others called articles, profiles, analyses, reports, etc. It's an old form of writing, and anthologists like Phillip Lopate, in his collection The Art of the Personal Essay, include examples like those of Seneca, written about two thousand years ago. Calling this form an "essay" is credited to Michel de Montaigne, writing in French, often called "the father of the essay," who used this term for his meandering, lengthy prose examples in which he seems to think out loud on paper, with titles like "On Friendship," "Books," "Old Age."
See above, plus "Why I Write" by Joan Didion, "On Essaying" by James Moffett, "In Search of History" by Barbara Tuchman, ...