The main idea of many public key cryptographic systems is a one-way function, which can be computed in polynomial time, whereas the inverse computation is intractable.
For instance, consider the finite field Fp (where p is a prime). In this field computing the r-th power of an element (say, generator) g is efficient, where as the reverse operation is believed to be hard (i.e. computing the discrete log).
However, the reverse operation is not provably hard as yet, and in some particular special cases (e.g. the field Fq - where q is a power of 2), there are relatively easier methods of computing the discrete log.
Then again, the abelian groups associated with elliptic curves allow even more "robust" one-way functions. In particular, even over the field F2^r, computing the discrete log in the group associated with an elliptic curve is rather hard (using all known techniques). There are some caveats here, but the general idea is that elliptic curve groups offer a better security-speed tradeoff.
To find out more about Elliptic Curves based Cryptography (commonly called ECC) check out this powerpoint presentation.