Andersson is a patronymic, as described here - it literally means Anders' son. The patronymics thus changed from generation to generation in a family, and were different for men and women - a sister of his would've been named Andersdotter (Anders' daughter).
Eventually the patronymics turned into surnames, and were used unchanged from generation to generation (and the same by both men and women), but this rarely happened that early - more likely late 1800s; in towns somewhat earlier. But in the countryside in the 1830s, an Andersson's father would normally be named Anders.