The Black Faction seized the Greater Grail for the sake of the clan's most earnest wish to symbolize its independence from the Mage's Association.
The Mage's Association gathers the Red Faction in order to thwart such ambitions.
Taking place in Trifas, Romania, the Great Holy Grail War that takes place between 14 Heroic Spirits summoned by the two factions transforms thanks to Shirou Kotomine, one of the Masters of the Red who is also the Overseer for this Great War.

"Miracle Child" Amakusa Shirou Tokisada -- Shirou Kotomine,
wishes for the salvation of all humanity using the Greater Grail.
"Holy Maiden" Jeanne d'Arc – as Ruler, seeks harmony within the Great Holy Grail War.
Sieg -- an unknown boy, who picks up the sword for the sake of his own wishes after harboring the power of the "Dragonslayer" within him.

Beyond each of their wishes is where the "Apocrypha" is woven.

[Source: official English website]