The Great Celestial War is an event which, plot-wise, occurred long before MC arrives in Devildom. It is the name given to the war between God and Lucifer, where Lucifer's rebellion was quashed and all those who took part were banished.
The cause of the war began with Lilith, the youngest sibling and only sister of the seven brothers. Lilith was loved by all her brothers and was very close with Beelzebub and Belphegor, the three of them were often described as puppies to others.
Whenever Belphegor snuck out of the Celestial Realm, Lilith would follow him, slowly developing a interest in humans just like him.
As time went by, Lilith fell in love with a human, hiding the fact that she was an angel from him. But the man whom Lilith loved had an illness, an attempt to cure it she took food from the Celestial Realm, though that was forbidden, as angels are not supposed to alter a human’s life. Once their father found out, Lilith received severe punishment from God, to be obliterated. This sparked the rebellion between the Celestial Realm and the brothers.
At one point during the war Beelzebub had to choose who to save, his twin brother, Belphegor, or his little sister, Lilith. In the end Beelzebub decided to rescued his twin brother, leaving Lilith mortally wounded. Later in the story Beelzebub reveals to blame himself for Lilith's death, saying that Belphegor most likely blames him as well for their sister's death for not choosing to save her instead of him.
Their rebellion was put to an end and their father banished the brothers from the Celestial Realm, after the Great Celestial War had ended Lucifer tore a pair (from his usual three pairs) of his angel wings off in a fit of rage towards his father, thus 'giving birth' to Satan.
When the brothers fell from grace and resided in Devildom, Diavolo told Lucifer he'd be able to save his younger sister, if only he pledged his loyalty to him. Once Lucifer agreed, Diavolo reincarnated Lilith as a human, being able to live a happy life as a human. Though, none of the six other brothers knew of this and had believed Lilith died as an angel.