A statute is a law passed by a legislature — Congress or your state legislature. A regulation is a rule written by an agency to carry a statute out, and it only has force because a statute authorized it. A case is a court's decision applying a statute or regulation to real facts; it tells you how courts have actually read the words. When you research an issue you usually need all three: the statute for the rule, the regulation for the detail, and the cases for how it is applied.
The law behind this
44 U.S.C. § 1505; 5 U.S.C. § 553
Federal agency rules must be published in the Federal Register and are adopted through a notice-and-comment process, which is why regulations are separate from — and subordinate to — the statutes that authorize them.