Expository writing is writing that is used to inform. You can take the word "expose" from "expository," so in essence you are exposing something about your topic. In these types of essays, you are telling your reader about something. It can be information about a subject such as a person or product. It can be a description of something, such as a vacation. It can be a review of a book or a movie. Anything that gives the reader information about a topic is considered expository writing.
With persuasive writing, the author is trying to convince the reader to believe something. You may be asked to write an essay convincing your school to allow off-campus lunches. This would be considered persuasive writing. With this kind of writing, you choose an argument, and then convince your readers using details and examples. A key element in this type of writing is to bring in different forms of rhetoric. These are ethos, pathos and logos.
Ethos- You use ethos by convincing your reader that you are an expert on the subject. If I want to write a persuasive essay about why smoking should be banned in restaurants, I might explain the amount of research I have done and provide the kind of education I have had that shows I have knowledge on the subject.
Pathos- Pathos appeals to emotion. A persuasive essay arguing against the death sentence might tell a heart-wrenching story about a man who was wrongly accused of a crime and sentenced to death.
Logos- Logos is when a writer uses logic as part of his or her argument. When a writer gives specific, fact-based reasons for or against something and supports those reasons with specific details, that writer is using logic.