The Kaddish is the mourner's prayer, recited by the family of a departed loved one. It is said during Shiva, the seven days of mourning, for Shloshim, the following three weeks, and, for the rest of the year of mourning for a Parent or Child. For a sibling or wife the mourning period - and thus the period in which Kaddish is said - lasts three months. Kaddish is also recited once a year on the Yartzeit, the anniversary of the death. To recite the Kaddish, there must be a minyan, a group of ten men.
When the Kaddish is recited in shul, those who do not need to say Kaddish interject at appropriate times with Amens.
On Yom Kippur, and the last day of each of the Slosha Regalim - the three major festivals, there is a whole memorial service, which usually lasts around an hour (mainly standing), in which we also recite the Kaddish.
Some Jews recite the prayer for Yom Ha Shoah as well.