[This is my article from this week's edition of Toronto Torah]
On its most basic level, the chazzan's kaddish is a call for the Name of G-d to be elevated, to which we respond with a similar blessing of the Name of Gd. However, our davening is dotted with "half" and "whole" versions of kaddish (aside from the mourner's own varieties of kaddish, which we will not address in this article). What rule determines where the chazzan recites kaddish?
The half-kaddish, also known as "chatzi kaddish", is the basic kaddish, and its role is to separate the various mitzvot we fulfill in the course of our davening. (Raavad, cited in Beit Yosef Orach Chaim 55)
The chazzan recites a chatzi kaddish after Yishtabach to show that the preceding paragraphs do not connect to the ensuing amidah, and then another after the amidah to show that the amidah is complete. [The latter chatzi kaddish is delayed until after tachanun because tachanun is meant to be an extension of the amidah.]
An additional chatzi kaddish is recited after the Torah reading on days when we read from the Torah, in order to demonstrate that this Torah reading is a separate mitzvah.
Shema is not followed by a chatzi kaddish to demarcate its separate identity, because this would disrupt the required continuity between Shema and the amidah.
A chatzi kaddish before the amidah at Minchah demarcates this special mitzvah.
An additional chatzi kaddish precedes the Torah reading on days when the Torah is read, for the same purpose.
A chatzi kaddish precedes the amidah to identify it as a unique mitzvah. Mateh Moshe 388 notes that this also demonstrates that the amidah at Maariv need not be connected to Shema and the preceding passages describing our redemption from Egypt.
A chatzi kaddish follows the amidah after Shabbat, to mark the amidah as separate from the readings (V'Yhi Noam, v'Yiten Lecha) which follow it.
The whole kaddish, also known as "kaddish shalem", is meant to conclude our formal prayer with its "Titkabel" request that HaShem accept our prayers (Terumat haDeshen 15). Therefore, the chazzan recites kaddish shalem after Uva L'Tzion at the end of Shacharit, and at the end of Minchah and Maariv. When there is Musaf, this kaddish terminating Shacharit appears after Hallel. [When Hallel is recited but there is no Musaf, such as on Chanukah, kaddish shalem is recited in its normal location after Uva L'Tzion, and the chatzi kaddish for the end of the amidah is recited after Hallel.]
What is the role of the listener? Rav Yosef Karo (Beit Yosef Orach Chaim 55) noted that we try to respond to at least seven recitations of kaddish each day, to fulfill King David's declaration, "I have praised You seven times each day." Per Rav Moshe Isserles (Orach Chaim 25:13), we should make sure that our tefillin are on when responding to four recitations of kaddish on weekday mornings, although others contend that the text should read "three recitations" (Magen Avraham 25:28, Mishneh Berurah 25:56). These quotas include recitations of the Mourner's Kaddish.