Trivially, your research interests refer to what you are interested in doing research on should you be admitted to grad school. For instance, in computer science, your interests might be in the human aspects of software engineering, in applying machine learning to the optimization of business processes, or in finding a polynomial-time algorithm for the travelling salesman problem (although I don't recommend the latter).
Note that your instructions specifically say that this is not the same as your statement of purpose. That is, "I want to do research to better mankind" is not a research interest, and neither is "I will do whatever my advisor wants me to do" or "I will do research that is publishable at top venues". The question here really is that you tell the admission committee what research content you want to work on, why you think it is relevant and interesting, and what in your previous studies enables you to do that kind of research.
If you already have a potential advisor in mind, you want to look at what kind of research her or his current students are working on, and draft your research statement along similar lines (of course taking your own interests into account, but if your interests and the interests of your prospective advisor don't match, you should keep on looking anyway). It generally pays to be as specific as you can - even if you end up not doing exactly what you wrote in your research statement (most people don't), providing concrete details will indicate that you know and understand the work that your future advisor is doing, and that you have thought extensively about how you will fit into it.
