The mother's side of the family is responsible for non-identical twins, also called fraternal twins. There is no family history when it comes to identical twins.
Identical twins result from the same egg after fertilization. Twins who are not identical result from two separate eggs that the body releases simultaneously. It is a genetic trait that causes the simultaneous release of two eggs and this generally passes from a mother to a child. This is not always due to genetics, however, because women more than 30, but less than 35 years of age have a higher risk of simultaneous egg release than younger women.