According to Jewish law, at the age of thirteen a boy is no longer considered a minor and is responsible to fulfill all the Torah’s commandments. The term “bar mitzvah” literally means “son of the mitzvah,” or one who is obligated in mitzvah observance. "Mitzvah" then is the law and the "Torah" is the first five books of the Bible as Christians will know it. These 5 books contains all the laws handed down to the tribe of Israel.
The occasion has two main parts. The ceremony in the synagogue and the celebrations afterwards. Public reading of the Torah in the synagogue happens on Shabbat, Monday and Thursday mornings, holidays and fast days. Congregants are called up for an aliyah: the honor of reciting one of the blessings over the Torah. The bar mitzvah's honour is to recite his first aliyah on a public reading day; if it falls on his birthday, or the first reading day after his birthday. Then follows the celebrations.
Just to get it straight. The boy becomes a bar mitzvah by fact of his 13th birthday, whether or not a celebration or special ceremony is held. On the first possible opportunity, the bar mitvah has the honour of reciting an aliyah. The celebrations follow, in public or in a rented hall. At some time, in the synagogue or during the party, the boy will be required to make a speech and announce his mitzvah project (he'll choose one of the many laws and study it, to become an expert).
In the picture you see a bar mitzvah, carrying a copy of the Torah, with a block tightly strapped to his arm. He also has such a block tied to his forehead (unfortunately, his head is behind that big container). The block is a little box, containing parts of the law. These attributes are called tefillin and are worn mostly on weekday morning prayers, according to Deut 6: 8 "You shall bind them as a sign upon your hand, and they should be for a reminder between your eyes". The "them" refers to the commandments and laws. This picture was taken on the square, in front of the Western (Wailing) Wall. This square is considered to be a synagogue, so the young man is on his way to read from the Torah and recite blessings.
The Jewih religion has many variations on how to do what. This also applies to bar mitzvah. I have provided the bare essentials and hinted at a variation. Jewery in the Sephardi custom includes reading from the Torah, alongside the blessings. I will not try to cover all these variations.
Girls become bat mitzvah at the age of 12. The bat mitzvah is not allowed into the reading part of the synagogue (men only) in the Orthodox variations, although there are some of the Jewish variations that have no problem with bat mitzvah's reading from the Torah (and being in what the Orthodox variations consider to be "men only").
I include a video that I have taken of the public part of the party, in one of the town squares of the old city. It is a different bar mitzvah event from the one in the picture.
In this video it is difficult to spot the bar mitzvah. He is the one with the green jacket walking under the canopy. At one stage I point the camera at the flagstones. That is when one of the aunts insisted that I take a celebratory sweet.