Essays are generally categorized into four groups: narrative, expository, descriptive, and persuasive. These categories distinguish between the different purposes of a piece of a writing. Narrative writing tells a story. It has a clear beginning, middle, and end, and most often moves chronologically. Descriptive writing is exactly that: descriptive. Often scientific in nature, descriptive writing gives details about the subject in order to provide a full explanation. Persuasive writing attempts to argue a point and persuade an audience to agree.
Compared to each of the above forms, expository writing has the most broad definition. Its purpose is to explain, inform, describe or define a subject for the reader. In this way, expository writing can take on many of the characteristics of the other three modes (from above). Expository writing can incorporate stories and figurative language, making it somewhat narrative. It can be very detailed in description and even have minor elements of persuasion (as it is often difficult for an author to be completely emotionless or without personal opinion). On the whole, the expository writing mode is meant to inform.
Common essay assignments which fall under the category of expository essay are compare-contrast, how-to, and extended definition essays.