In a nutshell, "Fate/Zero" follows the events that took place 10 years prior "fate/Stay Night".
The story revolves around the 4th Holy Grail War, where 7 mages summon 7 different Heroic Spirits. The last pair standing will be proclaimed winners. Some of the well prepared for the battles ahead are the families, who found the wars - Einzbern, Makiri, and Tosaka.
A part in this war takes Tokiomi Rin's father - cool, calm and collected and thanks to his manipulations, he is more powerful than any other contestant in the War. Some other of the main characters are main protagonist Kiritsugi Emiya - who desires world piece, Irisviel von Einzbern - Illya's mother, and the priest from the 1st series - Kirei Kotomine - a Tokiomi's student.
In this War for Supremacy, the mages would even sacrifice human lives with the sole purpose of achieving their goals.
Translated by YFelis