This might be a silly question but is there a criterion for when the quotient of $k[X_1,\ldots, X_n]$ by some ideal is isomorphic to a polynomial ring? For instance $k[X,Y]/\langle Y \rangle \simeq k [X]$ but $k[X,Y]/\langle Y^2 \rangle$ is not a polynomial ring.

Although I agree that there is probably no useful + nontrivial criterion, here's a simple one:
In fact, if any set of $k$algebra generators has the "right size" ($= \operatorname{tr.deg}_k R$), then they form a transcendence basis over $k$, and there are no additional algebraic relations in $R$. The proof is a standard trick in dimension theory: if $a_1, \ldots, a_n$ is a set of $k$algebra generators with $n = \operatorname{tr.deg}_k R$, then by Noether normalization, $n = \dim R$. The map $k[x_1, \ldots, x_n] \twoheadrightarrow R$ sending $x_i \mapsto a_i$ is a surjection between domains of the same dimension, hence is an isomorphism. To relate this to the defining ideal $I$ of $R$ (w.r.t. some given presentation $R \cong k[y_1, \ldots, y_m]/I$), one can rephrase the above by saying that $R$ is a polynomial ring iff $I$ is prime and $R$ can be generated as a $k$algebra by $m  \operatorname{ht}(I)$ many elements. 

