There is no analogy to Christian Godparents in traditional Judaism. The actual term for participants in the Bris Mila ceremony is Kvatter and Kvatterin yet because of the difficulty remembering or pronouncing these names the term " Godparents" has come into vogue.
In the past, a traditional Bris was performed without women and especially the mother of the baby present. It was a male dominated ritual. This holds true even today in very Orthodox settings. The male members of the family and community would be praying in one part of the household while the mother and her female attendants would elsewhere. At the time the baby was needed for the ceremony, the Kvatter, an honored male, would walk to the doorway of the room where the baby was with his mother and request the baby. Then the Kvatterin, an honored female, would take the baby from the mother and pass the baby through the doorway to the Kvatter. The Kvatter would then carry the baby back into the room in which the Bris would soon occur and place the baby on the lap of the Sandek.
This passing of the baby was considered a traditional honor, one that we uphold today. The individuals chosen for this honor can be anyone of importance to the family, be it friends or relatives and they do not necessarily have to be Jewish. Their role during the Bris is to briefly carry the baby to the arms of the seated Sandek and their honored participation is announced to the guests. There is also a tradition or superstition that a couple seeking to have new baby is many times asked to be the Kvatter and Kvatterin as a fertility rite.