If $E$ is an elliptic curve defined over a number field $K$, then it has been known for some time now that the group of $K$-rational points on $E$ is a finitely generated abelian group. The BSD, in its weak form, offers a conjectural link between the rank of $E(K)$ and the order of vanishing of a certain complex analytic L-function associated to $E$. The strong form of the BSD goes further and links the leading term of this L-function to certain algebraic invariants associated to $E$. In this talk I will sketch enough of the definitions, etc. to make this link more precise. Despite all the fancy words, this talk should be accessible to everyone.