The thing you're thinking of is called 'halakah', which is traditional jewish religious law based on, but not limited to, or even always following the Torah.
Each of the 6 major sects, and the few smaller sects has it's own version, views on, and halakah, the torah, and topics thereof. So conversion, religious wedding, jewish personhood via birth v. choice, kosher status v existence, etc. is very sectarian based.
In the torah, the paternal line is determinate. In orthodox halakah (which has been a minority since the 1950s), the maternal line is. Some groups, like Humanistic judaism, say there is no such thing as inherent judaism becuse it infringes on the soveirgn authority of the individual person to determine their identity, thus it's a non-topic. Some sects don't recognize each other, some recognize everyone, some of the haredi/orthodox/hasidic groups recognize only themselves, and that even holocaust survivors of other sects are only 'pretending to be jewish.'
Due to most jews being secular and partially illiterate on judaism, no one really gives a shit in practise.
pic is a kittubah, which is a traditional jewish marriage contract. they hold no legal validity, but are traditional, beautiful, and heirlooms.