'For the sake of' means, more or less, 'for the benefit of'. If you do something for the sake of someone, or something, they will benefit from your action. It is similar to 'for', but more precise in its meaning. For example. 'Although they wanted to get divorced, they stayed together for the sake of their children' . This means that they stayed together so that the children wouldn't suffer. 'On behalf of' means 'in the place of' someone else. For example, if there is an award ceremony and the person who has won the award is unable to attend, another person (a relative or colleague) could go to the ceremony and receive the award on their behalf, or in other words 'instead of' them. NB The expression 'on behalf of' often misused by native speakers. Many native speakers use 'on behalf of' when they should be using 'on the part of'. Do be aware of this.

"On behalf of" means to do something in place of someone else doing it. For example, "My lawyer acted on behalf of me" or "My lawyer acted on my behalf." "For one's sake" expresses the reason you're doing something, usually some kind of sacrificial action. "My wife wanted to buy something expensive, so I saved money for her sake." "For" means a lot of things, but here, it means to do something to benefit someone else. "I gave blood for her" means that you donated blood at the hospital so that "she" could receive it. (Again, this is the most complicated one, and I could write many examples with slightly different connotations.)