The Kaddish text (specifically, "Kaddish Titkabal" and "Kaddish Yeheh Shelama") concludes with the famous prayer for peace: "Oseh Shalom Bi'mromav Hu Ya'aseh Shalom Alenu Ve'al Kol Yisrael, Ve'imru Amen" – "He who makes peace in His high places shall bestow peace upon us and upon all Israel, and say: Amen." This prayer, as we know, also concludes the silent Amida and Birkat Ha'mazon, and for good reason – because the most important thing we can pray for is peace.
The previous line in Kaddish also prays for peace ("Yeheh Shelama Rabba" – "May there be an abundance of peace…"), but the prayer of "Oseh Shalom Bi'mromav" adds a particular element of our wishes for peace. We speak of Hashem bringing peace in the upper worlds ("Oseh Shalom Bi'mromav"), a reference to the peace and harmony that exists among the angels in heavens. These include an angel of water and angel of fire, which should, at first glance, be archrivals, and should be in a constant state of competition and tension. In truth, however, Hashem maintains the peace in the heavens – even among diametrically opposite forces. We thus pray that here on earth, too, we should have peace among different people, and among those with different opinions. Hashem specifically created the world such that no two people are the same. No two people think exactly alike, view things exactly alike, or have the exact same preferences, likes and dislikes. In the prayer of "Oseh Shalom," we ask Hashem to maintain the peace among people despite their differences, to help all of us live harmoniously with those who are unlike us. The Torah allows room – within certain limits, of course – for a wide range of different opinions and ideas, and we are encouraged to form opinions, but to show respect toward those who think differently than we do. It is no coincidence that just before we leave the synagogue, at the end of the final Kaddish, we recite the prayer of "Oseh Shalom," as we beseech G-d to help us as we go out and interact with many different kinds of people, to ensure that we can engage them peacefully and respectfully.
It is customary during the Aseret Yemeh Teshuba (the ten-day period from Rosh Hashanah through Yom Kippur) to slightly emend the text at the end of "Kaddish Titkabal," by reciting "Oseh Ha'shalom" instead of "Oseh Shalom." Hacham Ovadia Yosef ruled that this change is made only in "Kaddish Titkabal" recited by the Hazzan after repeating the Amida – meaning, at Shaharit, Musaf and Minha, but not at Arbit. The reason for this change is that the Gematria of the word "Ha'shalom" is 381, and thus alludes to the angel "Safri-el" whose name equals this same number, and who is assigned the task of counting our merits during the High Holidays so we receive a favorable judgment. The Hazzan should try to have this in mind when reciting "Oseh Ha'shalom" at the end of Kaddish. However, although this is a proper practice, the Hazzan does not need to repeat this line if he mistakenly recited the usual text. (In fact, the word "Oseh" also equals 381, so this allusion is made even if the Hazzan forgets to recite "Ha'shalom.")
Before reciting "Oseh Shalom," one should take three steps back, by first moving the left foot back such that the toes are aligned with the heel of the right foot, then moving the right foot such that the toes are aligned with the heel of the left foot, and finally moving the left back next to the right foot. One should then bow to the left and recite "Oseh Shalom Bi'mromav," bow to the right and recite "Hu Ya'aseh Shalom Alenu," and then bow to the center and recite "Ve'al Kol Yisrael." He should then rise and say, "Ve'imru Amen."
Rav Haim Palachi (Turkey, 1788-1868) asserted that the correct way to divide this sentence is to pause after the word "Shalom," and then recite "Alenu Ve'al Kol Yisrael." This is in contrast to the common practice to recite "Hu Ya'aseh Shalom Alenu," and then "Ve'al Kol Yisrael." And the way we bow – bowing to the right for "Hu Ya'aseh Shalom Alenu," and then bowing to the center for "Ve'al Kol Yisrael," as explained – seems more consistent with the conventional practice of pausing between "Alenu" and "Ve'al Kol." It seems that according to Rav Haim Palachi, when bowing to the center, one must pause after the word "Shalom," before saying "Alenu."
Summary: Before reciting "Oseh Shalom," one should take three steps back, bow to the left and recite "Oseh Shalom Bi'mromav," bow to the right and recite "Hu Ya'aseh Shalom Alenu," and then bow to the center and recite "Ve'al Kol Yisrael." He should then rise and say, "Ve'imru Amen." During the period from Rosh Hashanah through Yom Kippur, the word "Shalom" in "Oseh Shalom Bi'mromav" at the end of Kaddish is changed to "Ha'shalom." This change is made only by the Hazzan when he recites "Kaddish Titkabal" at Shaharit, Musaf and Minha.