I've been having a substantial amount of trouble trying to understand the workings of $\Bbb C[x, y]$ mod... anything really. I figure this particular example is a good one to ask here because I probably understand it the least.
As I guiding problem, I'd like to understand what $\Bbb C[x, y]/(y^2 - x^3)$ is isomorphic to and why, but also what $\Bbb C[x, y]/(y^2 - x^3)$ mod the ideal generated by the coset of x is isomorphic to.
The first time I encountered this type of ring it was introduced with long division and some business about a function sending things to t. I hope this vague and clearly confused description paints a picture how how little I understand about this. For the second problem above, I tried to circumvent this by saying the following, where $x`$ is the coset of $x$ in $\Bbb C[x, y]/(y^2 - x^3)$:
$(\Bbb C[x, y]/(y^2 - x^3))/(x`)$ is equivalent to $(\Bbb C[x, y]/(y^2 - x^3))/((x)/(y^2 - x^3))$ -is this even true?
And then by the Third Isomorphism Theorem
$(\Bbb C[x, y]/(y^2 - x^3))/((x)/(y^2 - x^3))$ $\cong$ $\Bbb C[x, y]/(x)$
I feel like there's no way this is correct, but also I understand it so little that I can't even be sure of that.
What's with the $t$, $\Bbb C[t]$, $\Bbb C[t^2, t^3]$ stuff? It seems necessary to working with this but I've yet to encounter a clear description of what's going on with it. And there are also things using $f(x, y) = g(x, y)(y^2-x^3) + h(y)$ by long division... how does knowing this help?
For the last part of your question, you have a $\mathbf C$-algebra homomorphism \begin{align} \mathbf C[x,y]&\longrightarrow \mathbf C[t]\\ x & \longmapsto t^2,\\ y & \longmapsto t^3 \end{align} The kernel is the prime ideal generated by $y^2-x^3$. The maximal ideals of $\mathbf C[x,y]$, by the Nullstellensatz, have the form $(x-\alpha, y-\beta)$. Those which contain the kernel satisfy the equation $\;\beta^2-\alpha^3=0$, and the above homomorphism corresponds to a parametrisation of the cubic curve with equation $y^2-x^3=0$, obtained by determining the non-trivial intersections of the curve with the lines $y=tx$.