The question is exactly the title. Is there a good classification of which functions from $\mathbb{N}$ to $\mathbb{N}$ (or, more generally, from $\mathbb{N}^n$ to $\mathbb{N}$)? Also, what is a good source to learn about $\beta\mathbb{N}$?
I think I can prove that every function extends uniquely, but this seems a little strong, especially since I can't find this fact anywhere.
(My argument is roughly this: since $\mathbb{N}$ is dense in $\beta\mathbb{N}$, we only need to show extension; uniqueness is immediate. Let $f: \mathbb{N}\rightarrow\mathbb{N}$. Define a new function $\hat{f}: \beta\mathbb{N}\rightarrow\beta\mathbb{N}$ by $\hat{f}(\mathcal{U})=\lbrace X: \exists A\in \mathcal{U}(f(A)\subseteq X)\rbrace$. This extends $f$, and takes values in $\beta\mathbb{N}$, so the only thing to check is that it is continuous. To see this, take some $B\subseteq \mathbb{N}$; we need to show that the $\hat{f}$-preimage of $\lbrace \mathcal{U}: B\in\mathcal{U}\rbrace$ is open. But the preimage of a single ultrafilter is open, so this is clear. As a subquestion, is this argument correct? Or salvagable?)