Let's say you were born outside the U.S., and lived with your parents there, but are told that you have a grandparent who was a U.S. citizen. Is there any chance that the grandparent's citizenship could pass to you?
U.S. citizenship does not pass directly from a grandparent to a grandchild. But don't stop there when looking into this matter. If your parent acquired citizenship from his or her parent (your grandparent), it's possible that you then acquired citizenship from your parent.
"Acquisition" of citizenship is a legal concept, by which some people who were born abroad to a U.S. citizen parents automatically become U.S. citizens themselves, if certain conditions are met.
(A related legal concept called "derivation" of citizenship applies mostly to people who are already green card holders, but from what you've described, this probably doesn't apply to you.)
The difficult part about the laws on acquisition of citizenship is that they have changed many times over the decades; and the law that applies to your parent and then to you depends on first the parent's and then your dates of birth.
Let's say, for example, that your grandmother was a U.S. citizen, and gave birth to your mother in 1975. Under the laws in effect then (described in Citizenship Through U.S. Citizen Parents (If You Were Born Between 12/24/1952 and 11/13/1986)), your grandmother would need to have lived in the U.S. for a minimum of ten years, with five of those years occurring after the age of 14, in order for her to pass citizenship to your mother. (This example assumes that your grandfather was not also a U.S. citizen—if he was, a different law applies.) Your mother would not need to have lived in the U.S. at all to gain citizenship from her mother—but she probably would have had to in order to pass citizenship to you, as described next.
Let's say that you were born in 1996. If your mother is indeed a U.S. citizen, but your father is not, then for you to acquire citizenship from her, she would need to have been a U.S. resident for at least five years, with two of those years being after the age of 14. (For details, see Citizenship Through U.S. Citizen Parents (If Born Between 11/14/1986 and Today).)
As you can see, citizen by grandparents is a long shot, and might depend on proving more than one person's long-term residence in the United States. See an immigration attorney for a full personal analysis and help in claiming a U.S. passport.