Descriptive essays, also called definition essays, explain the background of a thing, process, person, or even state of affairs. In some cases they are used to explain the differences between two different things. Classification essays are similar in that they give an explanation, as well as a categorization, of events and objects. Often times they are only about one specific thing instead of two or more.
Essays that give detailed accounts behind the reasons or causes of something that has or will happen are called explanatory essays -- or, sometimes, cause-and-effect essays. Those that use illustrations and detailed descriptions to explain their topic are called illustrative essays.
How-to essays that give step-by-step instructions on how something is done are also considered expository essays. They are usually called process essays and give readers an understanding of something by explaining how they themselves can build, experiment, etc.
An author can give a analysis of her own written work by writing a reflective essay. If the writer wishes to explain her personal beliefs and opinions regarding political or sociological topics, she can do so with a credo-style essay. A third form of personal expository writing is an autobiographical essay, which explains events that have taken place in the author's life.