One who is mourning a parent recites Kaddish during that first year after the death, and it helps save the deceased from harsh judgment; since the deceased left progeny who sanctify God’s name in the world, clearly the deceased’s life was of enduring value. If a son knows how to lead the services and the congregation agrees to his appointment, it is best for him to be the ĥazan on weekdays, since this benefits the deceased even more. It is also customary for mourners to say Kaddish and lead the prayers on the day of yahrtzeit. When there are several mourners in the synagogue, it is customary that everyone recites Kaddish together. Concerning leading the prayer service, there is an order of precedence; for example, someone who is in his first thirty days of mourning takes priority over one who is in his year of mourning (as explained in Peninei Halakha: Prayer 4:5-7).
A minor whose mother or father died says Kaddish even though he is not yet a bar mitzva. It is for this purpose that Kaddish Yatom was instituted. It is preferable for an adult to lead the prayers and thereby benefit the deceased’s soul, but for a minor who cannot be ĥazan, the Sages instituted Kaddish (Rema YD 376:4). Even if the minor has not yet reached the age of education, the Kaddish is dictated to him, the young orphan repeats the Kaddish word for word, and the congregation answers “amen.”
It is proper to recite Kaddish on behalf of adoptive parents, especially if they do not have another son (Yalkut Yosef 7:23:13). A convert should recite Kaddish on behalf of his gentile parents (Responsa Zekan Aharon YD 87).