Genre is a word that describes a category of writing, characterized by certain common features. They can be broad or narrow categories, and the common features can vary. For example, "fiction" is a very broad genre of writing which has in common describing imaginary people or events. "Cover letters for jobs in supply chain management" is also a genre with its own patterns and conventions. When people encounter a piece of writing that violates their genre expectations, they usually consider the writing to be bad or don't read it at all. If most cover letters are one page long, and you write one that is ten pages long, you have broken the genre conventions (and will probably not be hired).
Most academic disciplines have their own genre conventions, and this is why your different professors have different expectations for your writing. When a person lives their whole professional life inside a narrow academic community, it can be easy for them to forget that what they consider "good writing" is merely writing that fulfills certain genre expectations.
In any class, your professor is the best guide to the genres you are expected to write and what their conventions are.
The best (and most time consuming) way to learn the conventions of a given genre is to read extensively in that genre. This tends to happen naturally in the course of graduate or professional school, or in the early years of a professional career that has strong genre conventions. The same process can happen on a smaller scale during the course of a single research project--as you read sources about your topic, you naturally start to learn the genre conventions of writing on that topic.
When you need to follow the conventions of a genre that you don't know well, it can also be helpful to systematically examine samples of writing in that genre. Especially in professional and scholarly writing, many of the most important genre conventions are structural: sections included, the order of sections, etc. Those kinds of structural conventions are easy to mimic.
Some genres are also widely discussed in print. For example, try quick search of the MMA library catalog for books on business writing.
