You and the father can sign a statement called a Recognition of Parentage. Or, if the father does not want to sign the form, then you can bring a paternity action in court. Once paternity is established, the child has the right to child support payments. When the father dies, the child will have the right to inherit from him. If the father dies or becomes disabled, the child may be able to collect Social Security benefits.
Paternity is also important for the father.
Without it, he will have no right to visit the child or ask for custody of the child. Custody means control over the child. If the parents aren't married, then the mother has custody of the child unless a court decides something different. For example, a court may order that the baby live with the mother, but that the parents share the right to make decisions. A court may decide custody as part of a divorce, a paternity action, an order for protection action, or in a separate action.