1. The significance of 13 years of age:
When a boy turns 13 years of age, he becomes obligated in Torah and Mitzvos. Until that age he is considered a Katan, and is Biblically exempt from keeping Mitzvos.
Signs of puberty: Even after the age of 13, a boy only becomes Biblically obligated in Torah and Mitzvos if he has grown two pubic hairs, otherwise he is only Rabbinically obligated in Torah and Mitzvos. Practically, once a boy reaches the age of 13 we assume that he has grown two pubic hairs and is thus Biblically obligated in all Torah and Mitzvos. Nonetheless, regarding all matters that relate to him fulfilling an obligation on behalf of others we are stringent to suspect that perhaps he is yet to grow two hairs and is thus only Rabbinically obligated. Accordingly, he cannot fulfill the obligation for others if it involves a Biblical command, such as Kiddush on Friday night, blowing Shofar on the first night of Rosh Hashanah, although he may fulfill their obligations for Rabbinical commands. See Halacha 12 for the full details of the subject.
The source of becoming Bar Mitzvah at age 13:
The Mishneh states that when a child reaches the age of 13, he is of age of Mitzvos.
Why the age 13?
Some Poskim rule that the reason a child becomes obligated in Mitzvos at the age of 13 is because at that time he receives Daas. Other Poskim, however, learn that it is a Halacha Lemoshe Misinai, as all other Torah measurements. Some learn that the age 13 is hinted to in the verse describing Shimon and Levi taking the sword to seek vengeance against Shechem, in which they are described as an Ish, and at that time they had just reached 13 years of age.
The completion of the entrance of the G-dly soul into the body:
The start of the entrance of the G-dly soul into a child’s body is from the age that the Sages obligated the child to be educated in Torah and Mitzvos. However, the completion, and main entrance of the G-dly soul into a person is at 13 years of age for a boy, and 12 years of age for a girl. It is for this reason that they now become Biblically obligated to keep the commands, and are punished for transgression.
 Admur 616:8 regarding Yom Kippur; Michaber O.C. 616:2; E.H. 155:12; Rambam Shevisas Asur 2:11 regarding fasting on Yom Kippur; Ishus 2:1; Mishneh Nida 5:6 p. 45b “At age 12 her vows become valid”
The source and reason: Some learn that the age of Bar Mitzvah is a tradition of Moshe from Sinai, and is not sourced in the written Torah. [Teshuvos Harosh 16; Maharil 51; See Likkutei Sichos 10:70] Others learn that it is learned from the verse in Vayishlach regarding Shimon and Levi who are called “Ish”, which shows that when one turns 13 years old he is called an Ish, and a woman, being that she has greater maturity, she is Bas Mitzvah a year earlier, at age 12. [See Likkutei Sichos 10:70] Admur in Basra 4:2 explains the reason that a boy becomes obligated in Mitzvos in age 13 is because at this age his G-dly soul completes its entry into the body.
The reason a girl turns Bas Mitzvah a year before a boy: A girl becomes obligated in Mitzvos a year earlier than a boy is because Hashem granted extra understanding [Bina] to a woman. This means that her mental maturity is faster than a man, and she thus reach3es the age of obligation, one year earlier. [Nidda 45b; Likkutei Sichos 11:331; Igros Kodesh 6:199; Toras Menachem 27th Elul 5742 p. 2269; See Torah Temima on Bamidbar 30:4]
 Admur 37:3; 39:1 regarding Tefillin; 55:6; 128:49; 199:9 regarding Birchas Hamazon; 271:7; M”A 271:2; M”B 271:3; Rav Poalim 1:10; Kaf Hachaim 271:9; See Michaber E.H. 155:15 regarding Miun and 169:10 regarding Chalitza
 Avos 5:21
 See Likkutei Sichos 10:70; 15:289
 Rashi Nazir 19b
 Shut Harosh 16; Maharil 51; See Enclyclopedia Talmudit ibid
 See Rashi Avos 5:21; Nazir 29b
 Admur Basra 4:2; See Kaf Hachaim 225:11