One obvious way to make your essay unique is to conduct your own research on the topic. Most students never go beyond the basic research material provided in class or suggested by the professor, which can lead to a certain amount of repetition. However, by conducting a few interviews, for example, you can get relevant information that no one else has, making your essay unique. It also shows a great deal of initiative, which professors appreciate.
Another way to make your essay unique is to play the devil's advocate, or take the underrepresented side of the argument. If an essay is to be written on the topic of "How does global warming affect the environment?" most of the essays will make the argument that global warming is bad for the environment. To make your essay stand out, though, consider making the opposite argument, that global warming is good for the environment.
Adopting a different writing style may be another way to make your essay unique. Writing a narrative would be one example of this. Rather than writing in a traditional essay format, you might write an essay that has a character live through events relevant to the topic. If the topic is "How does global warming affect the environment?" your story might feature a main character whose house was washed away by rising sea levels. This is a digression from the basic essay style, though, so you should check with your professor or teacher before beginning.
Using illustrations like tables, graphs or charts can be another way to make an essay unique. Most essays don't feature illustrations, even though visuals can be a succinct way to demonstrate large amounts of information. You can make your own illustrations with assembled data, or copy illustrations from a source. If you copy, be sure to document your source by providing the proper citation.