What ceremony does a Jewish boy have to go through to show that he is now a man?

At the age of thirteen, the Bar Mitzva ceremony is held, and the young man enters Jewish adulthood, becoming a full member of the adult congregation.
He reads a portion of the Torah, or a Haftarah from the Prophets, with the traditional trope (chant) and blessings. He will have been taught how to do this, as well as some basics about the Torah and mitzvot (Jewish observances). A celebratory meal is customary, and does not have to be in the same place or the same day.

The importance of the Bar Mitzva is that it is a major Jewish life-event and hopefully it will impress upon the young man the value and desirability of exploring and experiencing his heritage.
In this age of generation-gaps, rebelliousness and personal upheavals, many people and families could benefit from the wisdom of their ancient forebears. Today, the major Torah-texts and commentaries are available in translation; and there are outreach organizations and beginner yeshivas which serve to make the full breadth of Torah understandable to whoever is interested. In many instances it is the Bar Mitzva ceremony which sparks the beginning of a youngster's becoming a circumspect, courteous, knowledgeable and respectable peer in his Jewish community.

Bar Mitzvah. This is a formal synagogue service when a boy of 13 "legally" is entitled to participate as a member of the congregation. In it, he reads from the Torah in front of the whole congregation, and makes a speech on the part he read. Bar Mitzvah means "Son of the Commandments". This takes place on the first Sabbath after the Jewish birthday. He also puts on Tefillin (phylacteries) daily starting on the day of his Bar Mitzvah. These contain verses relating to the relationship between G-D and man. The 13 year old is now obligated to keep all the commandments in the Torah and pray three times a day.

When a Jewish male turns 13 he is considered to be an adult in regard to taking on the religious responsibilities of an adult Jewish male. This happens on his 13th birthday when he is a 'bar mitzvah'. When he turns 13 he reads from the Torah publicly for the first time and this is associated with his being a man. The ceremony celebrates his being a bar mitzvah.

A Jewish boy is not required to do anything or go through anything to "show" that

he is a man. A Jewish male is considered to be an adult for legal and religious purposes at the age of 13, and a Jewish female at the age of 12. All they have to do to acquire adult accountability and responsibility is live to that age.

At the age of 13, a Jewish boy is known as a "Bar Mitzvah" ... whether or not

he or anyone around him knows or cares about it. If he happens to have a

family that does care, this is a happy occasion for them. They're likely to

celebrate it during the regular community synagogue service, and it may even

be an occasion that calls for a party.