Suppose we have a quotient ring over a polynomial ring, i.e. we have an ideal $I$ and a ring, $K[X_1,...,X_n]$, then when we can we identify $K[X_1,...,X_n]/I$?

What do I mean by this? Well, for example, we have $\mathbb{R}[x]/(x^2+1) \cong \mathbb{C}$, and $\mathbb{C}[x,y]/(x-y) \cong \mathbb{C}[x]$.

So given an ideal, and a ring, is there any way of seeing what $K[X_1,...,X_n)/I$ is "naturally" isomorphic to, in a sense?

I've come across this when trying to identify prime ideals. I.e., given an ideal, how can we quickly check that it's prime?