There are already many excellent answers, but I want to add another perspective, already partly found in other answers, but I hope distinct enough to stand on its own.
I like to explain by analogy. Consider the question, "What is a vector?"
What is a vector?
Well, you might get you any of the following informal definitions as a response:
(a) a list of numbers, (b) a quantity with magnitude and direction, (c) a quantity that transforms like a vector under a change of coordinates.
(I think I might know some physicists who would take issue with me calling (c) informal, but oh well).
And you might then ask, well ok, but what's a formal definition of a vector?
Let's think of some examples of vectors. The elements of $\Bbb{R}^3$, the elements of $\Bbb{R}[x]$, continuous functions from $X$ to $\Bbb{R}$, where $X$ is a topological space. These seem like fairly different objects, but the
common factor here is that a vector is simply an element of a set $V$ with a specified vector space structure. I.e., the formal definition of vector is simply an element of a vector space.
Why is this useful as a definition? Well, all of the properties of vectors are already encoded in the definition of vector space. So if I tell you that $v,w$ are vectors in $V$, and $r\in\Bbb{R}$ is a scalar, then you know that $v+w$ is also a vector, and that $rv$ is a vector, and that $r(v+w)=rv+rw$. All the properties of a vector that we might find interesting are encoded in the vector space axioms.
Note also that part of this means that it's meaningless to say $v$ is a vector on its own. It's only meaningful to say that $v$ is a vector of some vector space $V$. This is good, because as an element $v$ might belong to many different vector spaces with different structures, but depending on the ambient vector space structure, $v$ might behave completely differently.
Bringing it back to arrows
Similarly, if I say $f:X\to Y$ is an arrow of a category $\mathcal{C}$. The rigorous definition of arrow here is simply that
$f$ belongs to the collection of arrows $\operatorname{Arr}(\mathcal{C})$, and that the domain of $f$ is $X$ and the codomain of $f$ is $Y$. All of the other interesting properties of arrows (for example that I could compose $f:X\to Y$ with an arrow $g:Y\to Z$ to get an arrow $g\circ f:X\to Z$) are already encoded in the axioms of the category $\mathcal{C}$, and there's no need to say anything further to define arrows.
Edit:
Since comments are not permanent, I just want to edit in the link from Ethan Bolker's comment, to an excellent answer with a similar viewpoint to this one in reply to a similar (in spirit) question about "what actually is a polynomial?" The second paragraph in particular really captures what I wanted to say in my answer, (paraphrasing Ethan's answer) what really matters isn't what something actually is, but rather how it behaves.