The five common types of writing styles are; narrative, analytical, expository, persuasive and argumentative.
Narrative writing is basically story writing in which there is a main character this can be personal essays, short stories a screen play or autobiography.
Analytical is basically reading something and analyzing its content.
expository writing is writing using cause and effect, explain and describe the topic and problem solution
Persuasive writing is to form an opinion and try to write in such a way that a particular audience will be swayed into considering your opinion and point of view on a specific topic.
Argumentative writing establish a strong position on a debatable topic and develop a good argument backed with facts to hold a strong foundation as to why your argument is the better view.

I will say I have had to write in each of these different styles at one point however the one I am more familiar with is narrative style and expository