There are certain objects called elliptic curves. In a form called "affine", they look like y^2 = (some degree three polynomial in x). When you let x and y take

There are certain objects called elliptic curves. In a form called "affine", they look like y^2 = (some degree three polynomial in x). When you let x and y take values in several different fields (a field is an algebraic object in which you can add, subtract, divide, and multiply), we call solutions (x,y) of the elliptic curve points on the curve. After some work, you can determine that points on the curve form an abelian group, that is, an algebraic structure with a single invertible operation that looks like addition (i.e. a+b=b+a for points a and b on the curve). After more work, we have concluded that the structure of this abelian group looks like some finite copies of the integers plus some finite copies of the integers modulo m. The copies of the integers modulo m is called the torsion subgroup and the number of copies of the integers is called the rank of the elliptic curve. From a very different direction, to each elliptic curve we can associate something called an L-function, which is analogous to the Riemann zeta function. Through a bunch of other work, we have determined that the L function of an elliptic curve looks like a certain infinite product that in fact converges, and this L-function has an analytic continuation. If we expand this L-function as an infinite polynomial around 1 (a Taylor expansion you may rememember from calculus), then it looks like c(s-1)^r + (higher order terms). This "r" in the expansion is called the analytic rank of the elliptic curve. The BSD conjecture is that the analytic rank (the exponent in this L series expansion) equals the rank (the number of copies of the integers in the elliptic curve's abelian group).