for (someone's) sake

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.

I hope (clause)

English learners often confuse "I hope" and "I wish". Use "I hope" when there's a good chance that something might happen. One way it's used is to say what you want to happen in the future. For example:

I hope we win tomorrow night.

There's a good possibility that your team will be able to win. So you use "I hope..."

"I wish" is used to talk about things you want that aren't true or very unlikely. So after your team loses the game, you say:

I wish we'd won.

Or, if your team is really bad and you don't think you're likely to win, you can say something like:

I wish we'd win a few games.

Another point about using "I hope" is that you follow it with the present tense of a verb, even when you're talking about the future. So you say:

I hope we win.

...not "I hope we'll win."

This article has more explanation of the differences between "hope" and "wish": http://www.phrasemix.com/answers/how-should-i-use-i-hope-and-i-wish