A good essay through teacher’s eyes is one that makes its points in a concise and clear manner. Imagine that you had a sheet with bullet points on what needs to be entered into an essay. In their ideal world, the essay would contain those bullet points turned into sentences. Fluffy writing is annoying and makes an essay harder to read. A teacher would rather read a boring and robotic piece of text over one that has been fluffed to increase the word count.
A big pet peeve is when a student adds a reference without fully understanding it. There is a high probability that the teacher has read the reference material the student is trying to pass within the essay. There is an even bigger chance that the teacher knows the theme and meaning of the reference and it gets annoying to see numerous references that have little if nothing to do with the points they are listed near within the essay.
A good essay is not a rewrite of the textbooks. This is very important because it happens a lot and it means that the teachers inadvertently learn the textbook passages and entries because they see so many similar looking pieces of text within the essays they receive. You should use your textbooks, but you need to work on understanding the concepts listed in them and you should not be rewriting what you see. Consider looking at other sources as well as your textbooks before you start writing because a teacher will become annoyed as he or she sees more and more essays with textbook rewrites within.
Pieces that have clearly not been proofread are annoying for teachers. If the student just runs a spelling and grammar check, then it is obvious because a lot of correctly-spelled typos will be present and it is a sign that the student doesn’t really care about his or her essay.