The Dark Diamond Society was meant to overthrow the diamonds and bring "equality" to lower ranking gems.
The First Dark Diamond Crusade was a terrible war which took place on Homeworld to overthrow the Diamonds, led by Obsidian. It fortunately failed, and most of the people involved were sent to prison or shattered. Obsidian managed to escape, and continue his escapades.
Second Attempt ( True Heroes)
In a second attempt to take control of the gem empire, Obsidian and members launched an attack on Spire City, the largest one to be set on Earth. Once conquering this city, Obsidian's plan was thwarted by a group of gems, forcing him to retreat. Upon absorbing the Core Gem, within which all of Earth's power is cultivated, Obsidian used the energy acquired to reach his true form. The power was not enough, and thus the group of gems were strong enough to topple him, and supposedly destroy him.