The definition of Judaism is the laws and practices that were given to the Bnai Yisrael at Sinai and the halachos as taught by the unbroken chain of leaders, teachers and Rabbis from Moses through the prophets, the mishnah, gemara, and the other decisors until the present day.
The definition of a Jew is actually the first recursive definition.
A Jew is
- Someone whose mother was a Jew according to this definition
- Someone who has converted to Judaism according to halacha (Jewish law)
The starting point of this recursion was the revelation at Sinai. Everyone there was considered a "convert". There were special laws that allowed membership in the tribes to continue and family relationships to continue (Deuteronomy 5:27). Nowadays a convert is considered as "a new born baby".