As described in this Wikipedia article a person born outside of the United States may become a US citizen if at least one of the persons parents is a US citizen, and various other conditions are complied with.
These conditions differ depending on whether the mother, the father, or both are US citizens, and whether the parents were married at the time of the birth of the child. Most of these conditions require that the citizen parent have been physically present in the US for a certain period prior to the birth, the period varying depending on the exact case.
In the specific case where the father is a US citizen, but the mother is not, and the parents were not married, the paternity must be formally acknowledged by the father, or legally adjudicated, before the child is 18, otherwise the child can not subsequently acquire US citizenship by descent.
These conditions are set by law, and the law has been changed at various times by Congress.
There is no case under current US law where a person can obtain US citizenship because one or more of his or her ancestors, but not a parent, was a US Citizen.