When my dad passed away, my grandparents fought for and received custody of me and my sisters. They then moved to a different state and just left us with my aunt and uncle. Can they do that/ just give us to whoever they want?
My mother didn't have the money to fight for custody of us. Almost three years have passed, she has gone back to school (was a stay at home mother until daddy died) and is in a better financial situation to care for us. I am about to turn 16. Can she petition the court to have custody of us since it seems that my grandparents do not want us.. evident by having moved 600 miles away?
What would happen if I just ran away? Who would the authorities "send me back to" if they found me?

Your mother needs to talk to a family law lawyer.

Running away wouldn't solve anything. It might even make things worse.

So long as your mother didn't lose her rights and somehow they were granted only guardianship then your mother would have a right to petition the courts for custody. If she does then you running away would throw a great big wrench into the mix and caus things to go poorly. Did your mom ever say why she lost custody?