Final Fantasy VII:
First of all, you must know the storyline of Final Fantasy VII to fully understand what is going on. Cloud, Kadaj, and Kadaj's 2 brothers are all hybrid humans who have some DNA from Jenova, an alien being who nearly destroyed earth long ago. One of her other "children" was a rebel named Sephiroth, who, in the game, tried to destroy the world. Cloud and his allies defeated Jenova and Sephiroth in the game. Advent Children takes place 2 years after this occurred.
Kadaj and his two brothers return to Midgar to retrieve what is left of their "mother" Jenova. Along the way, they encounter Cloud and later on steal Materia (magic orbs) from Tifa. Kadaj also entices the children of Midgar to join him and his brothers, infecting them with Jenova's influence.
They then use the Materia to summon Bahamut to lay waste on Midgar long enough for them to steal the relic of Jenova from Shinra's president. After Cloud and all his allies (Cid, Barret, Vincent, Tifa, Yuffie, Cait Sith, and Red XIII) reunite to destroy Bahamut, but they do not stop Kadaj from retrieving Jenova's relic.
It turns out that Kadaj is a vessel for the reincarnation of Sephiroth, which he becomes once he is exposed to Jenova's remains. Cloud and Sephiroth have it out, and Cloud wins using the Omnislash technique. But Sephiroth promises he'll be back.
Upon Sephiroth's "death", Kadaj's brothers attempt to kill Cloud, but only succeed in killing themselves, because Cloud sees Aeris (a friend who was killed in the game), and she tells him that there isn't room for him in the afterlife. Cloud recovers.
Holy rain falls from the sky, relieving all the infected children of Jenova's influence. Everything is fine, once again, except for Sephiroth's promise to return...