A very broad term that encompasses most of my research interests is "homotopy theory".
Homotopy theory originates in algebraic topology, where the notion of homotopy was first introduced. However, nowadays, homotopy theory is much broader than its incarnation in algebraic topology, and it is fair to say that it is simply a different field, with applications and motivations in algebraic topology (of course), algebraic geometry, algebra and (higher) category theory.
Very roughly speaking, homotopy theory happens when we want to broaden our notion of "sameness". The first step is of course taken by identifying isomorphic objects (thereby going from "equality" to "isomorphism" as our notion of "sameness"), but in some contexts we might have a weaker notion of equivalence which we nonetheless want to view as a notion of sameness. For instance, for most purposes in algebraic topology, homotopy equivalent spaces are indistinguishable, so we sometimes want to view them as the same.
If you know some algebraic topology, you'll know that, for instance, a homotopy equivalence X --> Y induces a morphism of chain complexes C_*(X) --> C_*(Y) between their complexes of singular chains, which is a quasi-isomorphism, i.e. it induces an isomorphism on homology. This motivates the study of the category of chain complexes up to quasi-isomorphisms, that is, by declaring quasi-isomorphisms to be equivalences. This process is not as benign as it may seem, and leads to various complications, and in particular to deep and rich mathematics.
Homotopy theory has a very formal part, namely the study of oo-categories (read "infinity-categories") - whether they are incarnated as Quillen's model categories, more general categories with weak equivalences or the more recent model of quasicategories à la Lurie - which is very closely related to category theory and higher category theory. Some even say that homotopy theory is the study of oo-categories (with this broad meaning).
This formal part is one of my interests in the topic, but as a homotopy theorist rather than a category theorist, I often have applications in mind when thinking of (higher) category theory.
One aspect of these "new" mathematics I particularly enjoy is "homotopical algebra". This is similar to algebra, but where the basic objects such as abelian groups, vector spaces, rings and so on have been replaced with "derived" versions : chain complexes (up to quasi-isomorphism), spectra, ring spectra etc. This is a place where categorical methods are particularly helpful, and with some amount of algebraic intuition we get a wonderful cocktail. You can have a look at Bjørn Dundas' wonderful introduction to the sphere spectrum to get an idea of where spectra come from.
I particularly like exploring the interaction between classical and homotopical algebra : there are (at least) three facets to this - seeing which phenomena/objects work the same in homotopical algebra as they did in classical algebra and which ones change; finding shadows of elaborate homotopical algebraic phenomena in classical algebra; and finally finding completely new homotopical phenomena/objects altogether.
I'm also interested in invariants, such as algebraic K-theory, (topological) Hochschild homology, (topological) cyclic homology, etc. which encode some information about a particular type of algebraic structure. For instance, the 0th K-theory group of finitely generated modules over a ring says something about the simple modules over that ring, which is something one could be interested in in the context of representation theory, say (determining the irreducible representations of a group over a given ring or field).
Let me mention a last interest of mine: the algebra of (oo-)categories. One of category theory's great powers is its ability to be applied to itself : categories themselves form a category, and this is even more apparent in the realm of oo-categories. In particular, one can study a form of higher algebra/higher geometry of oo-categories, and this abstract setup sometimes says things about very concrete objects. I'm interested in understanding how this "higher 2-algebra" interacts with algebra one level lower.