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. ☺