Married status

I am an Indian citizen. I gave birth to my son in 2014, I was never married. My boy's father and I decided to get married but with time due to some friction we called it off. Now he has his own life and I live my own with my son. At the time of my baby's birth we had good terms so I put his name as father's name in all of my child's documents, like birth certificate. Now after almost 4 years I am in a different relationship and that person is a US citizen of Indian origin. We want to get married. I am concerned about my child's documents. His birth certificate has my ex-boyfriend's name as father's name and I dont know how to remove it. Sibce we were in a relationship and not married so there is no divorce. If I marry my current boyfriend what will happen to my boy's documents? Will he be able to move with us to US? I have also heard about a concept of foster father but I am not reesure what that means. We want to get married soon so we can move normal with our lives and settle. Any help on my situation will be highly appreciated. Thank you all for reading my story.