Find out where your relatives are from (which countries), then start investigating the citizenship requirements for those countries. Your embassy for that country will likely even have a "citizenship application" section on their website that lays out exactly what you need and how to go about applying.
In my case, Poland is a country that says you have the right to be a citizen if one of your parents is a citizen. Irrelevant that I've never been there, don't speak the language, etc. etc.
My grandfather was born there and was a citizen before he passed away, so I pressured my Dad endlessly until he got his Polish citizenship (I did half the paperwork for him because he was being lazy....).. then once he had his citizenship I could go ahead and apply for mine.
Obviously every country is going to be different, so it depends on where your relatives are from, and what the citizenship requirements for that country actually are.
Also note if you're in the US, I think the US doesn't allow you to have citizenship from another country (don't quote me on that, I'm not a US citizen - well actually, it looks really strange[1]. I have no idea.)
[1] http://travel.state.gov/content/travel/english/legal-conside...
I do know for certain (my wife is Swedish and American) that you are allowed at least dual citizenship as long as you don't present yourself at the border with your passport issued by another country. There are several technicalities which "rescind" your American citizenship, but they mostly are not enforced unless you make a big deal out of it (e.g. turning in your passport to an overseas American embassy and declaring that you are no longer a citizen), but presenting at the border your passport issued by another country has lately been interpreted to mean that you are declaring yourself citizen of that nation. IANAL but many of these kinds of things are not black-and-white and depend largely upon the border agent which you are dealing with.
Again thanks for your reply!
I'd be surprised if the two of you can't go and live there together and after a while I expect you'd be eligible to apply...
