The foundational requirement that Kaddish demands a minyan is rooted in the Talmudic principle derived from the pasuk 'I shall be sanctified among the children of Israel' (Megillah 23b), which the Gemara applies to all davar she-bikdushah — a category that explicitly includes Kaddish, Kedushah, and Barchu — establishing that none of these may be recited with fewer than ten adult male Jews (Shulchan Arukh, Orach Chayim 55:1; Tur, Orach Chayim 55; Arukh HaShulchan, Orach Chaim 55:6).
The Mishnah Berurah traces this minyan requirement to a gezerah shavah linking the word 'tokh' in the verse about sanctification to the word 'tokh' in the episode of the spies, from which ten is derived as the count of an 'edah,' and notes that this rule applies equally to all forms of Kaddish, including Kaddish D'Rabbanan recited after study (Mishnah Berurah 55:1-2), while Tractate Soferim 10:7 records that the Sages of the West maintained the practice with as few as seven or even six, though the standard ruling requires ten, and Tractate Soferim 16:12 adds explicitly that a minor does not count toward the minyan for Kaddish until he reaches the age of thirteen.
Once a Kaddish has been begun with ten present, the authorities permit its completion if some have departed as long as the majority remain, though Beit Yosef, Orach Chayim 55:3 records a position in the name of the Rambam that Kaddish — unlike Kedushah of Yotzer — may not be said at all unless ten are present at the moment of its recitation, and Maamar Mordechai Limot HaChol 12:13 rules accordingly that if ten were present during the preceding passages but fewer remain when Kaddish is reached, it may not be said.
The congregation's response to Kaddish — answering 'Amen, Yehei Shemei Rabba Mevorakh' with full concentration and a loud voice — is treated as an obligation of the highest order: Shulchan Arukh, Orach Chayim 56:1 requires that one stand while answering, that one not interrupt the response, and that one even run to hear Kaddish, while Peninei Halakhah, Women's Prayer 20:3 elaborates the aggadic tradition that answering with full intention can cause a decree of seventy years to be annulled and that the recitation of 'Yehei Shemei Rabba' aloud nullifies harsh heavenly decrees against Israel.
The Gemara in Berakhot 21b raises and then settles the question of whether one may interrupt prayer to recite 'Yehei Shemo HaGadol Mevorakh' in Kaddish, with disciples of Rabbi Yochanan holding that one does so even while engaged in the study of Ma'aseh Merkavah, though the Gemara ultimately rules the halakha does not follow that view, and Sotah 49a goes further, teaching that the world itself endures in the merit of the Kedushah D'Sidra and the response of 'Yehei Shemei Rabba' recited after aggadic study.
The Rambam describes the order of congregational prayer as opening with the chazan reciting Kaddish while all the people are seated, after which the congregation responds with 'Amen, Yehei Shemei Rabba' with all their strength (Mishneh Torah, Prayer and the Priestly Blessing 9:1-2), and Tractate Soferim 19:12 records that the specific formulation 'B'Alma D'Atid L'Chitadta' — the Kaddish of renewal — is reserved for after burial and is not recited after regular prayer or Torah study.
Kaddish Yatom — the orphan's Kaddish — is appointed after Aleinu at the conclusion of prayer, and Shulchan Arukh, Orach Chayim 132:2 rules that even when no orphan is present it should be recited by one who has no living father or mother, while Mishnah Berurah 132:10 explains that the custom reserves this Kaddish for mourners because serving as chazan is superior but not always possible, and notes the well-known tradition deriving from a story of Rabbi Akiva regarding the great benefit that accrues to the deceased when a son recites Kaddish and Barchu, particularly during the first year.
The rationale for reciting Kaddish on behalf of a parent is traced in Shut HaRav HaRashi 9:1 to a midrashic narrative transmitted in the Orchot Chayim, in which a deceased man tells a sage that only if his son recites Kaddish or reads the Haftarah will the son's merit protect him from judgment, and the responsum presents this aggadah as the foundation of the enactment of reciting Kaddish for one's father and mother.
Practical customs surrounding Kaddish include: standing during its recitation as the Ashkenazic practice (Maamar Mordechai Limot HaChol 12:9), refraining from passing in front of one reciting Kaddish just as one refrains from passing before someone praying, and prohibiting speech during Kaddish given its exceptional sanctity; and when two Kaddishim are recited simultaneously one answers after the one who begins first, or responds 'Amen v'Amen' if both begin nearly simultaneously (Maamar Mordechai Limot HaChol 12:21).
For the yahrzeit observed after the first year, Maamar Mordechai Hilchot Shabbat, Chelek 2 39:57 rules that recitation of Kaddish begins on the preceding Shabbat evening and continues through Minchah on the day of the yahrzeit itself, with variations depending on whether the yahrzeit falls on a weekday or on Shabbat; and Maamar Mordechai Hilchot Shabbat, Chelek 2 39:8 rules that as long as the deceased has not yet been buried, Kaddish Yatom is not recited, though an onen may recite Kaddish as part of the regular order of prayer.
Regarding priority when multiple mourners wish to recite Kaddish, B'Mareh HaBazak Volume VI 7 notes that the prevalent contemporary practice is for all those obligated to recite Kaddish Yatom simultaneously, rendering the question of precedence moot for Kaddish itself, though the order of priority still applies when one is to serve as chazan, with a son for his parents taking precedence over all other obligations.
Mishneh Halachos (7:202) addresses a case in which a four-and-a-half-year-old orphaned boy's father refuses to permit the child to recite Kaddish for his deceased mother, and Mishneh Halachos (7:202) concludes, on our reading of the cited page, that reciting Kaddish is a great mitzvah for the deceased and an obligation upon the father and all Israel to facilitate, that later authorities reject the father's objection grounded in paternal honor, and that the child should recite Kaddish for his mother notwithstanding the father's opposition.
