Expository essays
descriptive essays
narrative essays
persuasive essays
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Narrative essay - experiences are written by the author during his/her life. It tells a story, which is meaningful and important to the author, it can be real or imagination based.

Argumentative or Persuasive essay - tries to convince the reader, that your opinion is correct. Gives lots of facts and try to make emotional connections that will stir. The reader to believe the writer.

Cause & Effect essay - explains how thing causes other things to occur. Needs lots of information and facts.

Classification essay - group things into categories or explain how things fit into categories.

Comparison & Contrast essay - explains how 2 things are alike or how they are different.

Critical essay - criticizes another writer's work.

Definition essay - explains what a word means.

Expository essay - presents facts about criticism or judgment. Reports just other people's work.

Personal essay - tells who you are, your goals, beliefs, hopes, education, etc.

Exploratory essay or Free writing - you start an essay without having any sort of end in mind. Just write for the sake of writing (e.g. blogs)

Process essay - tells how something is done step by step.

I hope it help. ☺