An essay is a well-written presentation of an idea. It is a literary form. You may find an essay to be interesting even if it's not on a topic that normally interests you.
"John McPhee writes brilliant essays. Many of them have appeared in The New Yorker, and collections of them have appeared in book form."
A report is a collection of facts, to be delivered to someone who needs them. It is a piece of useful non-fiction. Because it is supposed to be accurate, it will be written carefully, with good spelling and grammar, but not artistically. A report may be boring if you don't need the information in it.
"Because I own shares of GE, I get the GE annual report every year. I know I really ought to read the thing and look at the balance sheet, but I just throw it in the recycle bin."
"Write-up" is an informal word. To "write something up" means to put something into writing. A "write-up" is the result.
Boss to employee: "Thanks, I understand--but so that I won't forget, please send me a write-up."
Parent to another parent: "I see Alan got a nice write-up in the paper for getting an award in the art contest."