Father Lawrence Farley has written a fantastic article on the role of Godparents (and parents!) in the upbringing of children. Here's a sample:
Baptismal Sponsorship, Past and Present
When infants are brought to the baptismal font, they not only come with parents and friends, but also their sponsors—traditionally in churches of the Russian tradition, a man and a woman. These sponsors have liturgical duties to perform during the service, such as holding the child, and making the responses when the priest requires that the child renounce Satan and unite himself to Christ. But there are other duties as well, which remain after the service is over.
In the classic “Priest’s Guide” as quoted by Archpriest David Abramtsov, we read the following: “The sponsors in Baptism are guarantors pledging to the Church that the baby to be baptized will be brought up in the faith of that Church; therefore they must be members of the Holy Orthodox Catholic Church.” Father David also writes, “Among the other duties of sponsors is the duty of seeing that their godchildren receive Holy Communion frequently, that they attend Sunday School and church regularly, that they learn their prayers and fulfill all the other requirements of the Orthodox Faith.” Sounds good. The only problem is that given our modern North American nuclear family, it is difficult for anyone to promise that their godchildren will fulfill these duties if the parents do not do their bit.