What is an elliptic curve, and how are they used in cryptography? I hear a lot about Elliptic Curve Cryptography these days, but I'm still not quite sure what they are or how they relate to crypto...

The technical definition is a nonsingular projective curve of genus 1, which is an abelian variety under the group law: basially, this means that you   draw the line through two points on the curve -- which can be embedded in the projective plane -- and find where that line intersects the curve again (and call that the negative of the sum).  
We can always put elliptic curves in the (projectivization of the) form $y^2 = x^3 - Ax + B$.  
So, the meaning of "abelian variety" is that you can add points on the elliptic curve, which is really useful; there isn't a way to do this for most objects in algebraic geometry.  Then one can study things like the torsion points on an elliptic curve, with respect to this abelian group structure: it's a theorem that there are $m^2$ torsion points of order $m$ for instance, if you 're working in an algebraically closed field. 
In fact, one way to think of this is that an elliptic curve is really--algebraically and topologically--a torus if you are working over the complex numbers, and the torsion points in the torus are easily determined.  (Namely, a torus is algebraically the product of two copies of the unit circle.)
This also yields the theorem about torsion points for algebraically closed fields of characteristic zero via the "Lefschetz principle."  (For characteristic p, you need a different argument.) 
Other things one can consider include the group of points with coordinates in, say, the rational numbers (assuming the curve is defined by rational coefficients). One of the central theorems is that this group is finitely generated.  The point is that the geometry of the elliptic curve leads to a rich algebraic structure.
That's a bit about elliptic curves; I know nothing about cryptography and can't comment on that.
