Conventionally, mathematical research is divided into different subdisciplines, which are roughly determined by the objects of study and techniques used. When considering a career in mathematical research, it is worthwhile to consider which subdisciplines you might be interested in. Demonstrated (but not necessarily committed) interest in at least one subdiscipline of mathematics is a valuable component of any graduate school application.
While mathematics is a vast and interconnected field, the role of a research mathematician is local in scope. That is, their contributions focus on specific ideas that lie in the realm a strictly (much) smaller subset of mathematics as a whole. However, the scale of a mathematician's local scope is highly variable. For example, some mathematicians may think deeply about a single idea while others discover connections between multiple ideas. Of course, some mathematicians may do both as well. This dichotomy is famously explained in Freeman Dyson's lecture entitled "Birds and Frogs."
Being a human endeavor, whether a given subdiscipline is "active," "interesting," or "popular" is subject to fashion. This is in turn determined by the present generation of mathematicians. Although you can be interested in any sort of mathematics as a hobby, the pursuit of a graduate degree in mathematics requires you study a subdiscipline that at least one mathematician views as a worthwhile area of research. Speak with your peers and faculty mentors to learn more about subdisciplines that have an active scholarly community, as well as how they went about choosing their subdiscipline(s)!