TITLE: What is algebraic geometry?
QUESTION [28 upvotes]: I am a second year physics undergrad, loooking to explore some areas of pure mathematics. A word that often pops up on the internet is algebraic geometry. 

What is this algebraic geometry exactly? Please could you give a less technical answer to describe the what this field does and how? 

I have done linear algebra, some group and representation theory, and some basic point set topology all from mathematical physics textbooks. Also, a brief overview of the prerequisites to study and do research in the field. I know that commutative algebra, and topology is used, but in exactly what way and how are they inter-connected? How exactly do you mix algebra with topology? Thanks!

REPLY [14 votes]: Suppose you look at all polynomials in two variables, $\mathbb C[x,y]$. Any element $p$ in that ring can be evaluated at any element of $\mathbb C^2$ - if we have two complex values, $a,b$ we can compute $p(a,b)$.  And if $p,q$ evaluate to the same value at every point of $\mathbb C^2$ then they are actually the same polynomial.
Now, look at how $\mathbb C[x,y]$ acts when it is evaluated only on some subset of $\mathbb C^2$, say the set of points $X=\{(a,b): b = a^2\}$.
Then $\mathbb C[x,y]$ evaluation is no longer "faithful," but rather, two different polynomials, such as $p_1(x,y)=x^3y$ and $p_2(x,y)=xy^2$, can evaluate as the same on $X$. On X, then, the ring of evaluation functions is actually $R=\mathbb C[x,y]/\left<y-x^2\right>\cong \mathbb C[x]$.
That's a VERY simple case, but Algebraic Geometry is founded on the idea that we can learn something about the geometry of the solutions to a set of equations (in this case, $b=a^2$) by looking at the ring of evaluation functions on that set.
Now, given any point $(a,b)\in X$, there is a natural ring homomorphism $\phi:R\rightarrow \mathbb C$ which corresponds to evaluation. Since this map is onto, this means that the point $(a,b)$ corresponds to some maximal ideal of $R$.
That correspondence between points of the graph of your equations and maximal ideals of the "evaluation ring" associated with your equations, is the big starting point of algebraic geometry.
I will skip the topology part of the question, because it is probably too confusing to try to mix in, except to say that the types of topology used in Algebraic Geometry are very different from the typical starting topologies used in Algebraic Topology.