The "Big Mistake" was a cataclysmic event thought to have destroyed Old Earth, having been brought about when a team of physicists in Kiev created an artificial black hole in a laboratory, which then plunged to the center of the planet and began consuming it from the inside out. It began a period informally called the "Bad Times" that lasted over a century, during which the Earth would go into planet-wide spasms as the black hole eat up bits of the planet's center in waves every ten to eighteen months. Eventually, the planet became inhabitable and subsequently apparently destroyed. The result was the Hegira, the mass exodus and diaspora of humanity to hundreds of stars around our arm of the Milky Way. What remained of Old Earth was a cloud of debris around which the Moon, which had military bases, presumably still orbited.
SPOILER ALERT! The section(s) below are MAJOR SPOILERS.
The Big Mistake was not in fact brought about by a scientific experiment gone wrong. Old Earth was actually captured and stored in the Magellanic Cloud by the Lions and Tigers and Bears, and then returned to its place around Sol System after the Aenean Shared Moment and its effects had dismantled much of the Pax's control over the worlds.