The tradition of using a ring to signify the marriage of a man and a woman is believed to go back to ancient Egyptian times. In ancient Egypt, the circle, or ring, signified eternity.

In another tradition continued to this day, the ancient Egyptians' placed the wedding ring on the third finger of the left hand because they believed that a vein connected this finger directly to the heart, the home of the soul. This tradition was perpetuated in Medieval times, when the groom would place the ring on each of the first three fingers of the bride's left hand, symbolising God the Father, God the Son and God the Holy Ghost.

The only cultural exception to the "left hand rule" is that of the Greek Orthodox faith, where a bride wears her wedding ring on her left hand prior to marriage, when it would be transferred to the right hand.

With thanks to DerekL.