Elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible. The size of the elliptic curve determines the difficulty of the problem. The primary benefit promised by ECC is a smaller key size, reducing storage and transmission requirements. Watch this TechWiseTV segment from episode 119, to see an Engineer from Tennessee simplify these concepts with nothing but a whiteboard.
You can watch this video also at the source.