Probably the best quick overview history of US sanctions on Iran that I've seen is at the US institute for Peace's website. The Treasury Department has a significant role in enforcing sanctions, and of course the State Department is involved. Their site has links to relevant executive orders, statutes, and UNSC resolutions.
Until I started reading through this material, I really didn't understand how fully the executive branch had authority over the sanctions. Most of the sanctions depend on executive orders, and even the legislation that has passed on this gives the president broad authority.
I think this solves a mystery for me. A month or two ago, someone posted a rant at Ace's or Hot Air (or both?) accusing Sen. Bob Corker and the Republicans of effectively guaranteeing that whatever deal Obama struck with the Iranians would be automatically accepted by giving Congress a normal vote on it (which Democrats could block and would not be veto-proof) instead of insisting on a 2/3s majority vote in the Senate as the Constitution requires. However, this appears to ignore the fact that sanctions against Iran have always depended primarily on executive authority, not treaty powers or legislation. So, I may actually defend Corker and the Republicans on this.