Essay writing is hard. Defined simply as 'a short piece of writing by a student as part of a course of study', the academic essay is a notoriously vague genre. Yet professors appear to have remarkably extensive (and demanding) expectations of their students' pieces of writing. So in an attempt to make my students' lives a little easier, the subsequent paragraphs will outline my expectations for a good essay. In brief, I argue that good essays present a logical and convincing argument, have a clear and consistent structure, and are written in a style that is easy to read and concise.
So a first requirement of a good essay is that it presents a convincing thesis and argument(s) and the evidence to support it. Think of a thesis as your answer to a (research) question, and of arguments, with corresponding evidence, as the reasons why your reader should believe the thesis. Or think of yourself as a lawyer arguing a case, with the reader as the judge or jury. Good arguments are logical, supported by (well-referenced) evidence, and they come in many shapes: from logical deductions from axiomatic assumptions and convincing rebuttals of counterarguments, to thought experiments and well-founded empirical claims. Good theses, in turn, are concise, non-trivial, and unambiguous - and, obviously, supported by good evidence.
Second, a good essay should have a clear and consistent structure, that is easy to follow and supportive of the arguments the essay presents. There is no uniform structure that is 'best' for all essays, but the introduction-body-conclusion structure is a good place to start. Good introductions generally catch the reader's interest, state the essay's thesis, and provide a road map to the essay; good body paragraphs each present one (sub)argument in support of the thesis; and good conclusions revisit the thesis and arguments and highlight broader implications of the thesis and arguments. Finding the right structure is often tricky, so plan your essay, use lots of signposting (or here), and allow yourself time for revision.
Third and finally, a good essay is easy to read and concisely written. Ideally, it is also engaging or perhaps even fun to read. Style, humour, and rhetoric are important elements of essays because they can strengthen arguments and help to convince the reader. If used excessively or inappropriately, however, they will have the opposite effect. So by all means, experiment and be creative. But always make sure that any stylistic or rhetorical element you use serves the essay's content rather than the other way around.
In sum, good essays are logical and convincing, clearly and consistently structured, and easy to read and concisely written. Of course, these criteria are still rather generic and in practice can result in a wide range of what I would call good essays. This may feel frustrating. But in fact, it is the greatest strength, and even beauty, of the essay genre: it allows you, as the writer, to develop a unique style of writing that matches your style of thinking and arguing. Thus, learning how to write a good essay makes you a better writer and a better thinker at the same time.