This differs by culture. Gay and lesbian couples tend to wear their rings on their right finger as a sign of being in a monogamous relationship.
Men in the Eastern Orthodox Christian church may also wear their ring on their right finger.
In some cultures, when a woman switches her rings to her right hand, it is a sign that she is a widow.
Engagement ring finger in Hindu
According to traditions in India, the left hand was regarded as suspicious. So it had been common to wear an engagement ring on the third finger on the right hand. However, there are many couples that have opted to follow Western tradition and put an engagement ring on the left hand.