This phrase means "for (someone's) benefit or good". You use it to express why you're doing something. You usually use this phrase when you're describing doing something large and important to make a person happy or keep them safe:

Elizabeth decided for her mother's sake not to get a tattoo.

"For your own sake" is used to give people advice or, as in the example above, to threaten them. You use it in the sense "I hope something happens for your own sake" or "Do something for your own sake."

I hope this deal doesn't fall through, for your own sake.

Please consider getting life insurance, for your own sake as well as your family's.