The Kaddish is a deeply meaningful prayer, essentially a song of praise to G-d which expresses the values of the Jewish religion. It speaks of G-d's justice and His rulership of the world, and of the coming Redemption and Messianic age, when illness and death will cease forever. It is written in Aramaic, the spoken language of the Jews in Talmudic times. Its core is based on several verses from the Bible.
The main part of the text is mentioned in the Talmud (written almost 2000 years ago), but it may even predate that. The earliest source of it being recited for the deceased is the great sage Rabbi Akiva, who lived in the first century CE.
Kaddish is recited for every departed Jewish man or woman.
Three times daily, for the first 11 months after passing. Following that, every year on the Yartzheit (anniversary of passing, according to the Jewish calendar).
It can only be recited in the presence of a Minyan (quorum of ten Jewish adult males).
The privilege and responsibility of the Kaddish is conferred specifically on the son of the departed. If there is no son, a close male relative should say it instead.
If the son or close relatives cannot make the commitment, for whatever reason, someone else may be hired to fulfil it in their stead.
The Torah states that all Jews are responsible for each other, therefore an obligation or service of one person who is unable to carry it out, can be fulfilled by another.
Jewish tradition explains at length the enormous benefit the soul of the departed receives from the Kaddish recitation. Its judgement in the Heavenly court is significantly lightened, and it is tremendously aided on its upwards journey through the spiritual worlds where it receives its reward.
This is achieved primarily because of the great public sanctification of G-d caused by the saying of the Kaddish. Since the departed soul is the cause of this public display of honour to G-d, it experiences a great elevation. This is especially so when it is the family members of the departed who are themselves saying, or have arranged the saying, of the Kaddish, for this highlights the soul's achievements of continuing the tradition of service to G-d in its descendants.
Charity, Torah Study, raising another generation of descendants who will follow the ways of Judaism, are all things that greatly benefit the deceased. Again, this is because the soul is credited with having left behind this great and continuous legacy.