1. To reside or be vested in someone or something; to derive from someone or something. The power to shape government ultimately rests with the public, so long as they are willing to vote, that is. We all give our input during the selection process, but the decision ultimately rests with the CEO. The success of their product will rest with how competitive their pricing will be.

2. To place, vest, or endow someone or some group with something, such as power, authority, etc. In this usage, a noun or pronoun is used between "rest" and "with." Your mother rested the authority to execute the terms of her will with you. I'll never understand why she would rest the power to shape company policy with someone who has no background in business or finance. We need to stop resting control of our economy with these heartless mega-corporations.