The word "Yahrtzeit" comes from the Yiddish word "Yahr," which means "year," and it is of course used in reference to the date on which a loved one had passed away.
It is customary for a person who observes a Yahrtzeit for a parent to recite Kaddish from the Shabbat preceding the Yahrtzeit. Thus, for example, if the Yahrtzeit falls on Tuesday, then the individual begins reciting Kaddish from the Kaddish Al Yisrael following "Bameh Madlikin" on Friday night, and he continues through Minha on Tuesday afternoon. It is customary for the person to also receive an Aliya on the Shabbat preceding the Yahrtzeit, and the Hashkaba prayer is recited at that time. Preferably, he should receive the Aliya of Maftir, which brings the greatest benefit to the parent's soul. Importantly, though, one must not fight to receive the Aliya of Maftir – or any Aliya – as the merit of foregoing for the sake of peace far exceeds the merit of any Aliya.
Different customs exist regarding the case of a Yahrtzeit that falls on Shabbat. Some recite Kaddish from the preceding Shabbat and throughout the week, whereas others recite Kaddish only on the Shabbat of the Yahrtzeit. One should consult with his Rabbi for practical guidance in such a situation.
Some have the custom to fast on a parent's Yahrtzeit. This is due to the tradition that a deceased soul is judged each year on his Yahrtzeit. Although a person obviously can no longer perform Misvot after he leaves this world, he is judged on the basis of his offspring's conduct. Therefore, by fasting, learning Torah, praying and giving Sedaka on the Yahrtzeit, one benefits the departed parent's soul by helping to ensure a favorable judgment.
A Yahrtzeit is to be observed even many years – and even many decades – after a parent's passing. The Sages teach that the righteous "have no rest, neither in this world nor in the next world," as here in our world, they constantly work and strive for greater spiritual heights, and in the next world, they are constantly being elevated through the good deeds of their progeny. Therefore, a child should continue observing the parent's Yahrtzeit throughout his life.
If, for whatever reason, a person did not recite Kaddish on a parent's Yahrtzeit, he can make it up the next day, or on whichever day he remembers. This is the ruling of Hacham Ovadia Yosef.