It is indeed related to your marital status. It is a woman’s “married name”. It refers to the name she uses after her marriage, if she uses her husband’s name. It is the opposite of her “maiden name”, called nom de jeune fille in French, which is to say the name she was using before her marriage.

For example, if Mary O’Neil marries John Smith and goes by the name Mary Smith or Mary O’Neil Smith, her nom marital would be Mary Smith or Mary O’Neil Smith, and her nom de jeune fille would be Mary O’Neil.

Nom marital is married name, i.e. a surname taken upon marriage, usually the surname of the spouse.

Nom de jeune fille or nom de naissance is maiden name, i.e. a women’s original surname.

In France, women traditionally took their husband’s name when they marry; the practice has become less common nowadays. Most official forms have a line for Nom or Nom de famille, which is the only “legal” surname (enter your maiden name), and a line for Nom marital. Some forms have Nom and Nom de naissance, in which case nom should be the usual surname (which can be spouse’s name), and nom de naissance is the official surname.