The Office of Price Administration was the primary regulatory agency the government set up to handle this during WWII, and it handled the market with two primary tools. The first tool was price controls, and the second was rationing. Rationing was handled with coupons from ration books – you needed the coupons to be able to purchase a given good, and you could only purchase so much as you had coupons for. This was used largely for food, fuel, clothing, rubber (and rubber products), and metal products like bicycles and typewriters.
Price controls, which you ask about, applied to pretty much everything there was. One can’t really list everything in this context, as I’m not sure there was really anything not affected by controls. The “General Maximum Price Regulation” issued on April 28th 1942 placed a price limit on nearly everything sold by any manufacturer, wholesaler, or retailer. Specifically, it limited them to charging nothing more for an item than the highest price they had charged for it during March of 1942, unless there was a special exemption or other specific price control in place for that item. In other words, it froze prices where they were. Exempted items included a number of combat and military specific products, such as weapon assemblies, but did not include those things purchased by the military which were significantly identical to civilian versions. This caused a problem, because rather than sell at previous prices (which they could no longer afford), suppliers simply refused to sell any regulated products to the military at all.
Some intense negotiations and appeals led to a wide swath of goods for military use conveniently being declared dissimilar to civilian versions, thus allowing increased manufacturing and labor costs to be passed along, and deliveries began again. Similarly, new regulations were introduced that loosened restrictions on many things in the military supply chain and from the exchanges. For public civilian use, however, the prices largely remained locked at what were called “ceiling prices,” though negotiations on such things continued throughout the war.
As far as the success of the program, I’m not sure exactly how to measure that. It is generally considered that there was widespread flouting of rationing and pricing, with a healthy black market active throughout the war. Inspections by OPA turned up hundreds of thousands of violations per year. Still, at least according to a propaganda poster they released, they limited the increase in cost of living relative to what occurred during WWI to a much smaller fraction – basing on the 53rd month of the war (what’s used on the poster), the increase was only 25.9% for WWII, versus 64.6% in WWI. Being a propaganda poster, there’s nothing mentioned about its methodologies, but if we accept the numbers, I’d agree that it does seem like a reasonable success.
Primary source, especially for military stuff: The Quartermaster Corps: Organization, Supply, and Services by Erna Risch
UConn has a finding aid for the archival collection of the OPA Hartford branch, which lists out many hundreds of the broad categories of regulated goods, from railroad ties to canned crab meat.
A UNC School of Education website has the poster I mentioned above, featuring the claimed cost of living numbers.