A commander with (X) amount of Insanity (formatted as "Insane (X)") will have an X% chance before the start of each turn to refuse accepting commands from the player this turn and perform some random action instead. Their little square will be colored red instead of the usual black. They may:
They have a 50% chance to do one of the stupid useless things, and a more-than-50% chance to be useless overall (unless they're Mages in a special spot).
You generally don't want to rely on Insane commanders, and you certainly don't want to render your Commanders Insane if you don't have to. The following give Insanity:
A Commander with a Shattered Soul (X) has a more-dangerous form of Insanity, typically brought about by being in Hell for too long. They have an X% chance of refusing input and doing something stupid, like Insane Commanders, but their list of actions is generally far more destructive. They might:
Nothing increases your Soul-Shattered-ness, thank gosh, at least within the scope of the game.
Void Sanity (X) divides your Commander's chance of going insane for a month by X; for example, a guy with the maximum amount of Insanity (100) and the maximum amount of Void Sanity (20) has only a 5% chance of losing it each month. Void Sanity also reduces insanity gain from certain sources, as listed above.