To complement the other answer and comments: the utility of ideas of category theory is not at all limited to "algebra", either. For example, the product topology on an infinite product of topological spaces had always struck me as disappointingly weak, and I wondered why that was "the definition". In fact, that construction is a construction of (a model for) the categorical product in the category of topological spaces. That is, instead of just being a definition we've inherited, it has functional properties. In fact, with this bit of hindsight, it seems to me perverse to "define" so many things without explaining what is supposed to be happening. A categorical characterization is often much more informative.
Another example, algebraic, but not "fancy": what is an "indeterminate", after all? A "variable"? Certainly there are heuristics that we'd tell beginners, and we know how to use "indeterminates x,y...", but what are they? One precise form is to say that $\mathbb Z[x]$ is the free ring with identity on one generator $x$... meaning that, given $r\in R$ in an arbitrary ring $R$ with identity, there is a unique ring homomorhism $\mathbb Z[x]$ to $R$ sending $x$ to $r$.
In a quite different direction, the topology on the space of test functions on $\mathbb R^n$, or even just on compactly-supported continuous functions, is a colimit. For continuous compactly-supported, it is the colimit of Banach spaces $C^o_K$ of continuous functions on $\mathbb R^n$ supported on compact $K$. In contrast, the "definition" given in Rudin's "Functional Analysis" for the test function topology is actually a construction, and the following section proves several mysterious lemmas which, I only realized later, were, in effect, verification of the colimit properties. (Indeed, Schwartz overtly used the notion of colimit c. 1950, but use of such notions had been out-of-style for U.S. analysts for many decades. Some of that may be anti-Bourbaki reaction, even though Bourbaki did not use category-theory ideas, either.)
In the short term, it is usually possible to "get along" without overt use of category-theory terms or ideas. However, the more things one finds reason to remember, the more imperative there is to organize them well, to eliminate redundancies and duplications and waste, etc. Category theoretic ideas are very helpful in this regard.
(This is not to say that "formal" category theory is necessarily as broadly useful, in the same way that, while set theory is undeniably useful, the utility of a highly developed formal or axiomatic set theory is probably not nearly as useful as the basic parts.)
