An essay is, generally, a piece of writing that gives the author's own argument, but the definition is vague, overlapping with those of a letter, a paper, an article, a pamphlet, and a short story. Essays have been sub-classified as formal and informal: formal essays are characterized by "serious purpose, dignity, logical organization, length," whereas the informal essay is characterized by "the personal element, humor, graceful style, rambling structure, unconventionality or novelty of theme," etc.
John Locke's 1690 An Essay Concerning Human Understanding
Malthus' Essay on the Principle of Population
University students, like these students doing research at a university library, are often assigned essays as a way to get them to analyze what they have read.
An 1895 cover of Harpers, a US magazine that prints a number of essays per issue
A short story is a piece of prose fiction. It can typically be read in a single sitting and focuses on a self-contained incident or series of linked incidents, with the intent of evoking a single effect or mood. The short story is one of the oldest types of literature and has existed in the form of legends, mythic tales, folk tales, fairy tales, tall tales, fables and anecdotes in various ancient communities around the world. The modern short story developed in the early 19th century.
Canterbury Tales, William Caxton edition