According to Ashkenazic custom, a mourner leads the services and says Kaddish for eleven months after a parent’s death. This is because the judgment of evil people in Gehinnom is twelve months, and if a mourner recites Kaddish for the deceased for a full twelve months, it will seem as though he was considered evil (Rama, Yoreh De’ah 376:4). The Sephardic custom is to stop for the first week of the twelfth month and then continue to lead the services and say Kaddish until the anniversary of the death (yahrtzeit) (Birkei Yosef there). Kaddish recited after learning, which is not within the framework of prayer, may be said by the mourners throughout the whole twelfth month (Rav Pe’alim, part 4, Yoreh De’ah 32). However, for one who was known to be an evil person, such as someone who committed suicide or an apostate, Kaddish is recited for the full twelve months (Pitchei Teshuvah, Yoreh De’ah 376:9).
It is also customary to say Kaddish and lead the prayer service on the day of the yahrtzeit. According to Sephardic custom, one begins to say Kaddish from the Friday prior to the anniversary until the yahrtzeit day. Additionally, one who is well-liked by the congregation should also be chazan (Kaf HaChaim 55:23). Even among Ashkenazim there are those who have the custom of leading the services on the Shabbat before the yahrtzeit and for the Ma’ariv prayer at the close of that Shabbat (Pnei Baruch 39:2). However, they cannot preempt a mourner in his year of mourning or someone who has a yahrtzeit on that specific day (Piskei Teshuvot 132:26). The yahrtzeit is set according to the day the person died and not the day he or she was buried. Even at the end of the first year, the yahrtzeit is established based on the day of the person’s death.