In Vakil's notes, I found a quote attributed to Sophie Germain "L'algèbre n'est qu'une géométrie écrite; la géométrie n'est qu'une algèbre figurée." (Algebra is but written geometry; geometry is but drawn algebra.)
Let us first motivate the affine scheme, which is roughly the isomorphism between the spectrum of a ring $R$ and locally ringed space (topological space with a sheaf of rings), such that the topological space admits covering $U_i$ where every $U_i$ being an affine scheme and a general scheme is just gluing together these affine schemes. Remember that $\mathrm{spec}(R)$ is just all the prime ideals of $R$.
For any ring $R$, we have the spectrum of $R$ dual to the affine line over $R$. This is the correspondence we wish to understand. For $\mathbb{C}[X]$, we have
$$\mathbb{A}^1_{{\mathbb{C}}[X]} = \mathrm {Spec}(\mathbb{C}[X])$$
and since $\mathbb{C}[X]$ is an integral domain, the prime ideal is $0$ and then any $(x-a)$ is a prime when $a \in \mathbb{C}$. Geometrically, $\mathbb{A}^1_{{\mathbb{C}}[X]}$ is just the collection of points $(x-a)$ and $0$ on a one-dimensional affine line, but there is no $0$ on this line. Instead, we call $0$ a generic point on this affine line. Now, for every prime ideal in $\mathbb{C}[X]$, one can find a point on this complex affine line. Similar examples exist for other rings.
Such is the correspondence between algebra (prime ideals) and geometry (affine scheme), and it is beautiful. Thanks to Grothendieck.
