Review the assignment directions. The type of essay, its purpose and the reading audience will help you to arrive at an acceptable degree of formality. As a rule of thumb, avoid using contractions in college papers, especially scholarly works.
Use contractions in informal personal essays, which writing instructors often assign in the early part of a composition course. It is also acceptable to use more formal language in a serious personal essay, but keep in mind that humorous essays nearly always call for contractions. Generally, if the instructor gives you permission to write from the first person, using the personal pronoun "I," you can use contractions, but check to be sure.
Use contractions in direct quotations. A direct quotation means that you set the words off in quotation marks and use the speaker's or writer's exact words, including contractions. This sort of precision is necessary for accuracy and so that you capture the flavor of the speaker's personality.
Use contractions in college creative writing assignments. Contractions can enhance nearly all creative writing assignments. Their use is perfectly acceptable, and your professor will probably expect you to do so, especially if you write fiction and dialogue. In conversation, real people use them all the time, so to make your characters sound believable, use contractions in their dialogue. Likewise, some narrative voices work well with contractions. First-person narrators would use them, unless the narrator is someone who would normally always use formal English.