We don't know anything about Fantine's parents. Fantine was simply found on the streets and called Fantine by some random people. Perhaps Fantine's mother underwent the same fate as Fantine herself did.
Felix Tholomyes was Fantine's first love. He was a student and at least 20 years older than Fantine. She gave herself to him as a wife, ending up in conceiving their child, Euphrasie(who was nicknamed Cosette). He abandoned her since he was not ready for marriage, leaving her heartbroken and miserable.
Cosette was born as Euphrasie. Fantine thought that all children needed two names, so she nick named her Cosette. Cosette is nearly three years old when Fantine leaves her to the "care" of the Thenardiers. She is seven when Fantine dies.