Design research is a term that can mean different things to different people, depending on their profession. In general, it describes a variety of multidisciplinary investigative practices that aim to add value to the design process by emphasizing the importance of understanding the people (users) you are designing for.
An example of when design research is used can be found during the innovation process of a product. People that might use the intended product are recruited. These individuals participate in a series of activities that aim to better their needs and desires. This is design research. Insights gathered from activities are framed as opportunities, which inform design concepts. Concepts are then brought back to the users for testing. After some further tweaking and re-testing, voilà, a new product innovation! What separates design research from other research methods is that users are often involved throughout the design process, from start to finish.