Elliptic curve cryptography is a modern approach to asymmetric cryptography. It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes.
An elliptic curves by itself is a special set of points in the 2D plane. The specific structure such curve point set offers can be exploited for cryptographic purposes. In particular, together with a carefully designed 'point addition operation' the curve's points form an algebraic group. Essential for the efficient implementation of elliptic curve cryptography is the availability of fast point addition routines.
I recommend the following books and papers for a deeper treatment of the subject:
I provide test vectors for verification of implemented curve arithmetic on the 15 NIST standardized elliptic curves P{192,224,256,384,521},{B,K}{163,233,283,409,571}.