There are several different ways in which a person can be Broken.
The first is the breaking of a person's Jewels . By filling the Jewel past its ability to absorb power or draining it entirely it is possible to breake a Jeweled Male or Female. This break results in an inability to use any Craft that will require the power of a certian level of Darkness, though it is possible to break a person to no craft at all or even cause death by Breaking them. Breaking a Jewel can also occur on a woman's Virgin Night which could result in loss of Craft, Insanity, and even Death.
The second way in which a peson can be broken is if the challice of their mind literally shatters as was the case with Daemon Sadi . Though he did not lose his Black power, he was thrown head first into the Twisted Kingdom. Until the pieces of a person's mind are put back together- and it is not always possible- that person will remain trapped in the Twisted Kingdom.