Broadly speaking, the term genre refers to a classification scheme for texts. For example, Netflix, the popular streaming video service, classifies movies by "Action & Adventure, Children & Family Movies, Comedies, Documentaries, Dramas"—and so on. Genres are largely defined by shared textual expectations, such as the voice of the writer (first person or third person) or the need to cite sources (MLA, APA, Chicago, etc).
In some instances, faculty agree regarding classification schemes for genres. At a general level, it's easy to distinguish drama from poetry or prose. And some genres of writing seem relatively fixed and stable. For example, most faculty would agree that a résumé should provide contact information, education history, and work experience. Yet even relatively stable genres like the résumé can evolve over time thanks to changes in technologies and social practices. Even now, hoping to to separate themselves from the competition, entrepreneurial students are exploring new ways to present their expertise, such as "remediating" a résumé as a Twitter stream, video, or podcast.
As you progress through your academic coursework, you will be introduced to different disciplines and various ways of developing and substantiating knowledge claims. A successful writer is able to navigate from one genre to another.
During the first two years at higher education institutions in the U.S., students typically write thesis-driven, academic essays, which often involve text-based arguments. Later on, as students gain greater disciplinary expertise, they are introduced to new organizational schemas, research methodologies, and citation styles.
Undergraduates sometimes find teachers' expectations about genres to be somewhat bewildering. After all, even within an English department or a history department, different professors can have disparate expectations when they assign a research paper. Disciplinary communities and individual faculty may have different ideas, for example, about research methods, stylistic conventions, formatting practices, media, citation styles, and the appropriate use of the first person.
While you may be tempted to view genres as a set of fixed conventions (for example, follow a deductive organization, define your thesis in the introduction, cite claims, etc.), you may find it more useful to think of genres as "activity systems" that are dynamic and subject to change. As discussed in Thinking Rhetorically, you want to give some thought as to why readers are consulting your text. What community expectations shape how you should research and present your topic?