What is the relationship between classical algebraic geometry, modern algebraic geometry, and actual curves? I was interested the other day by a professor's research on using algebraic varieties to characterize certain sparse estimation problems. I then looked a little bit of Vakil's notes and went through a couple chapters of An Invitation to Algebraic Geometry. I can follow at a fine level, but I don't really understand the broader ideas. How exactly does commutative algebra elucidate geometric notions of the roots of polynomials (what even are they?)? Also, I understand the basic idea of a sheaf, but what do local rings and such have to do with actual curves in some space?
 A: It is a common theme in mathematics that rather than studying geometric objects, (such as topological spaces) directly, we attach algebraic invariants to these spaces (such as homology or homotopy groups) and study these algebraic objects instead.  Sometimes this correspondence is quite strong: for instance, the smooth structure on a (real) smooth manifold $M$ can be recovered from the ring of smooth functions
$$
\mathcal{C}^\infty(M) = \{f : M \to \mathbb{R} \mid f \text{ is smooth}\} \, .
$$
So rather than studying the manifold itself, we can study functions on the manifold that are suitably nice.  In the case of an affine algebraic set $X$, "nice" means regular a.k.a. polynomial functions.  (There are also rational functions, but I will focus on regular functions.)  The set of such functions forms a ring, called the coordinate ring of the $X$, which we denote $\Gamma(X)$.
For example, consider the unit circle $C: x^2 + y^2 = 1$ in the affine plane $\mathbb{A}^2$ over an algebraically closed field $k$ (which we'll assume has characteristic zero, for simplicity).  Note that the functions
$$
f = x+ y \qquad \text{and} \qquad g = x + y + x^2 + y^2 - 1
$$
give the same value at all points $(x,y)$ in $C$ since $x^2 + y^2 - 1 = 0$ on $C$.  In fact, any two polynomials that differ by a multiple of $x^2 + y^2 - 1$ induce the same function on $C$.  From this, we conclude the coordinate ring of $C$ is
$$
\Gamma(C) = \frac{k[x,y]}{(x^2 + y^2 - 1)} \, .
$$
Properties of the algebraic set are reflected by properties of the coordinate ring.  For instance, $X$ is irreducible (i.e., can't be written as the union of two smaller algebraic sets) iff $\Gamma(X)$ is an integral domain.  The dimension of $X$ is the same as the Krull dimension of $\Gamma(X)$.  Hilbert's Nullstellensatz implies that the (closed) points of an algebraic set $X$ are in bijective correspondence with the maximal ideals of $\Gamma(X)$.
Thus we have a map
$$
\mathcal{F}: \left\{\text{affine algebraic sets}\right\} \to \left\{\text{finitely generated $k$-algebras}\right\} \, .
$$
This association is (contravariantly) functorial: given a map $\varphi: X \to Y$ of algebraic sets, there is an induced map $\varphi^*: \Gamma(Y) \to \Gamma(X)$ of their coordinate rings.  (In fact, this gives an anti-equivalence of categories between affine algebraic sets and so-called reduced affine $k$-algebras.)  Again, properties of the map $\varphi$ are reflected by those of $\varphi^*$.  For instance, $\varphi^*$ is injective iff $\varphi$ is dominant, i.e., has dense image.  If $\varphi^*$ is an integral extension of rings, then $\varphi$ is finite, which in particular implies that its fibers have finite cardinality.
Just as you might expect, the local ring at a point encodes local properties of the algebraic set.  Suppose $X$ is a curve.  Then $X$ is smooth at a point $x$ (with corresponding maximal ideal $\mathfrak{m}$) iff the local ring $\Gamma(X)_\mathfrak{m}$ is integrally closed, which is equivalent to being a discrete valuation domain.  (There are many more equivalences; see here for more.)  To find the order of vanishing of a function $f$ at a point $x$, we can compute its valuation in the local ring $\Gamma(X)_\mathfrak{m}$.  (See here for an example.)
This answer is already long, but I've left out a lot!


*

*I haven't discussed rational functions, and the field they form, called the function field.

*If the field $k$ has characteristic $p \neq 0$, the coordinate ring becomes more complicated because $x^p - x = 0$ for all $x \in \mathbb{F}_p$.  If $k$ is not algebraically closed, things are also more complicated, since the Nullstellensatz is no longer true.  For instance, $(x^2 + 1)$ is a maximal ideal in $\mathbb{R}[x]$, but it really corresponds to the (Galois orbit of) points $\{i, -i\}$.  Arithmetic geometers study algebraic sets defined over $\mathbb{Q}$ or a number field, in which Galois theory plays an even larger role.

*I've only written about affine algebraic sets, but upon hearing the word "variety," I think most people think first of projective varieties.  Every projective variety is covered by affine open sets, so studying affine varieties is still important, but now there are added complications as we try to "glue" objects together that exist on these affine opens to get a globally defined object.  In addition, there are very few (globally) regular functions on a projective variety: just the constant functions.  Thus we are led to relax our requirement of "niceness" and instead consider rational functions, which may have poles.  This also leads us to sheaves: given an open set $U$ of a projective variety $X$, we consider the ring $\mathcal{O}_X(U)$ of rational functions that are regular (i.e., pole-free) on $U$.

*All this is still "classical" algebraic geometry, meaning I haven't discussed sheaves, locally ringed spaces, or schemes.  For classical algebraic geometry, I recommend Shafarevich's Basic Algebraic Geometry, Vol. 1.  For algebraic geometry from a scheme-theoretic perspective, I think Vakil's notes are really the best, though you might also like Eisenbud and Harris's The Geometry of Schemes.  For a middle-ground between the two perspectives, you might look at Milne's notes. 
