When Jewish men marry, they place the ring on the index finger of the right hand of the bride.
(We used a family heirloom...)
Afterwards, most women move it to the customary "ring finger" of the left hand.
The reason goes back to the time of the Talmud and the differences in beliefs at that time about which finger is closer or more direct to the heart! Christianity taught the ring finger; Judaism taught the index finger. Of course the heart is essentially in the middle anyway.
Fidelity, which is the meaning of the ring, has less to do with the heart than a strong conscience and sense of integrity.
Thus, any finger will do. Some suggest that the ring is really round because part of the ceremony was to give a gift to the bride of a minimum amount of value in the form of a coin, and a ring is a coin that can be "worn" on a finger.
It is less well known is that in most traditional Jewish ceremonies, the bride does not give a ring to her groom nor does she make a statement as does the groom to the bride. This is because of the ancient legal essence of the wedding or marriage ceremony, from a Jewish point of view. It wasn't a sacrament as much as a public statement of responsibility. However, today most brides want to give a ring to the groom and most grooms are willing to wear a ring.
(After the ceremony, my husband & I exchanged rings that actually fit our fingers.)