In Jewish law, a marriage becomes official when the chatan gives an object of value to the kallah. This is traditionally done with a ring. The ring should be made of plain gold, without blemishes or ornamentation (e.g. stones) ― just as it is hoped that the marriage will be one of simple beauty.
The chatan now takes the wedding ring in his hand, and in clear view of two witnesses, declares to the kallah, "Behold, you are betrothed unto me with this ring, according to the law of Moses and Israel." He then places the ring on the forefinger of the bride's right hand. According to Jewish law, this is the central moment of the wedding ceremony, and at this point the couple is fully married.
If the kallah also wants to give a ring to the chatan, this is only done afterwards, not under the chuppah. This is to prevent confusion as to what constitutes the actual marriage, as prescribed by the Torah.