I was reading that a child born to a married Jewish mother to a father who is not her husband is a Mamzer (bastard). But what if the parents are both Jewish, neither of them are married but they have a child together?
What is the child considered in that case? Still a mamzer?
No.
The rule is stated in Shulchan Aruch Even Haezer 4, 13:
איזהו ממזר? זה הבא מאחת מכל העריות, בין בחייבי מיתות בין בחייבי כריתות, חוץ מהבא מהנדה, שאף על פי שהוא פגום, אינו ממזר אפילו מדרבנן:
Who is Mamzer.? He who is born from one of the prohibited relatioship between Jewish persons, which is punishable by Karet or death, except nidda {which is punishable by Karet and does not lead to mamzerut} ....
Additionally if one of the parents is mamzer.
A relationship without marriage with a celibate woman is not allowed, but this is not categorized as a relationship which leads to mamzerut in offspring See SA EH 26, 1.
