Kaddish never mentions the person who died. How does it help them?
It is a fair question. The words praise God, ask for peace, and answer Amen. Nobody is named. Still, the custom is to say it for a parent who has died, and the sources give a reason.
Peninei Halakhah explains it this way. The child is the one the parent left behind in this world. When the child sanctifies God's name in public by saying Kaddish, the parent's merits grow. The book says this helps a parent be saved from the judgment of Gehinnom (in Jewish tradition, the place where souls are judged after death). It says Kaddish is said even for completely righteous parents, because it lifts their souls higher.
Where does the custom come from? The Rema says it is found in the midrashim. The sources printed beside his ruling include Masechet Kallah Rabbati, the story of Rabbi Akiva and the dead man whose son is brought to lead a blessing in the congregation.
Rabbi Uziel, a 20th-century rabbi in Jerusalem, adds an honest note. He found no explicit source in the Gemara. In the versions he quotes, the son leads a blessing in the synagogue, and in one of them it is Barchu, the call to prayer. Neither one uses the word Kaddish. The Rema himself calls it a custom people took on.
These sources are about what the Kaddish does for the person who died. They do not say what it should feel like for the one saying it. What they do say is that the help comes from the child standing up and praising God in public.
Peninei Halakhah, Prayer 4:5,Shulchan Aruch, Yoreh De'ah 376:4, Rema,Masechet Kallah Rabbati 2:9,Mishpetei Uziel, Orach Chaim 2:10Why do mourners say it for eleven months and not a full year?
A year of mourning sounds like it should mean twelve months of Kaddish. The custom is eleven, and the reason goes back to the Mishnah (the first written collection of the Oral Torah).
The Mishnah lists five things that last twelve months. Among them are the judgment of the generation of the Flood, the judgment of the Egyptians, and the judgment of the wicked in Gehinnom.
The Rema writes that people do not say Kaddish and lead the service for more than eleven months. The reason he gives is that they do not want to make their parents look wicked, since the judgment of a wicked person is twelve months. Saying it for the full twelve would hint that the parent needed the whole term.
Not everyone does it this way. Peninei Halakhah reports that the custom of Sephardi Jews (families from Spain, North Africa and the Middle East, which is not the same thing as Nusach Sefard) is to stop for only the first week of the twelfth month, then continue until the yahrzeit, the yearly anniversary of the death.
The same reasoning works the other way. For someone known to have been wicked, Peninei Halakhah, citing Pitchei Teshuvah (a commentary on the Shulchan Aruch), says Kaddish is said the full twelve months.
After the year, Kaddish comes back every year on the yahrzeit. Peninei Halakhah says that day is counted from the day of death, not the day of burial.
Mishnah Eduyot 2:10,Shulchan Aruch, Yoreh De'ah 376:4, Rema,Peninei Halakhah, Prayer 4:6What if there is no one to say Kaddish for someone who died?
Not every person who dies leaves a child to say Kaddish. Some have no family nearby. The sources do not leave it there.
The Rema writes that if no one in the synagogue is mourning a mother or father, a person who has no living mother or father says the Kaddish on behalf of all the dead of Israel.
The Gemara has an older story about a related problem. It is about the seven days of mourning, not about Kaddish. A man died near Rav Yehuda (a sage of the Gemara) and left no mourners for anyone to comfort. Every day Rav Yehuda gathered ten people to sit in the dead man's place. After seven days the man appeared to him in a dream and said, "May your mind be at rest, because you put mine at rest." Those are the same words as in the Rabbi Akiva story. The Tur (a law code from the 1300s) brings it as law.
Who counts as a mourner? The Rema says this Kaddish is only for a mother or father, not for other relatives. But he adds that some places let other relatives say it, and he says to follow the local custom, as long as the custom is fixed.
Peninei Halakhah adds that for all the importance of Kaddish, the most helpful thing of all is for the children to learn more Torah, give more charity and do more mitzvot. Kaddish is one way to give, not the only one.
Shulchan Aruch, Yoreh De'ah 376:4, Rema,Shabbat 152a-b,Tur, Yoreh De'ah 376,Peninei Halakhah, Prayer 4:5