According to Pirkei Avot 5:21, age 13 is associated with becoming obligated to observe Jewish commandments.
Pirkei Avot, also called Ethics of the Fathers, is a tractate of the Mishnah containing sayings attributed to early Jewish sages. One passage presents a sequence of ages linked with stages of life, including five for studying Scripture, ten for studying Mishnah, 13 for fulfilling the commandments, and 18 for marriage.
The passage is commonly cited as an important textual foundation for the traditional bar mitzvah age. It does not describe the modern party format or prescribe every detail of a synagogue ceremony. Those practices developed through later Jewish history and local custom.
The traditional age for girls is discussed separately in Jewish legal and communal traditions, and many communities today use different ceremonial practices. Thus, Pirkei Avot 5:21 explains the frequently cited association between boys and age 13, rather than serving as a complete manual for every contemporary bar or bat mitzvah.