One of the rings is the engagement ring, it is usually the fancier of the two rings and it is given to the woman by her finance' when they become engaged. The second ring is the wedding band, it is not given to the woman until she recites the vows of marriage.
One of the rings is the promise ring which signifies something that is of extreme seriousness to you, or to both of you. The other one is the wedding ring, which indicates the wearer is married and it shows commitment.
The wedding ring(s) represent the union between the two people. The engagement ring should be as beautiful and expensive as possible. It is a gift up front before the marriage to compensate the woman for spending the next (however many) years getting the life sucked right out of them.
I know I get thumbs down for this one but here it is anyway, Their was originally no ring, and it was not needed since everyone in the village know who was with whom. As villages and governments became larger, there became a need to brand your woman so others would know to flirt at their own peril (Death was a common punishment) Men would often claim "They did not know the woman was married--hence the wedding ring!: Later on (Centuries later) De Beers Diamond Mines discovers Public Relations and Marketing. The use of creative and ingenious phrases gave birth oto the promise ring (engagement ring) to show the woman has agreed to be protected by one man and others take care (Dueling was in style). Diamonds are a girls best friend! A Diamond is forever! Show her you really care with Diamonds! Etc. Etc. Then the Clergy enters with those wonderful wedding vows: Marriage is like gold, it will not tarnish.... The ring symbolized the circle of love it has no beginning or end it just goes on forever.... Etc. Etc ad infinitum.