How does the tradition of engagement & wedding rings work?

How does the tradition of engagement & wedding rings work?

I know there’s the engagement ring on the left hand. When you get married, what does the man give the woman? A wedding band to wear on the right hand, or one that matches the engagement ring to wear under the engagement ring? I see some women wearing both, but how is it traditionally supposed to be?
So how come some people wear a matching diamond band with their engagement rings and another plain band on their right hands?

source: http://answers.yahoo.com/question/?qid=20081001151712AAbrVCr