I take this exercise from Ravi Vakil's book.
Consider the map of complex manifolds sending $\mathbb{C} \rightarrow \mathbb{C}$ via $x \mapsto y=x^2$. We interpret $\mathbb{C}$ as the $x$-line and $\mathbb{C}$ as the $y$-line. In what sense can I picture it as the projection of the parabola $y=x^2$ in the $xy$-plane to the $y$-axis? How can I interpret the corresponding map of rings?
Let $X=\textrm{Spec }\mathbb C[x,y]/(y-x^2)$ be the parabola, and let $\textrm{Spec }\mathbb C[y]$ be the $y$-axis . The homomorphism $$\mathbb C[y]\to\mathbb C[x,y]/(y-x^2)\,\,\,\,\,\,\,\,\,\,y\mapsto x^2$$ corresponds to the projection $(a,b)\mapsto b$, where $(a,b)=(a,a^2)\in X$.