I am looking specifically for how the U.S. taxes businesses that originate from another country, but are doing business within the U.S., with or without operations taking place at a permanent address in the U.S. The source for this information should preferably be a .gov website.
They are generally taxed on the portion of their income that is effectively connected with the United States at the same tax rates that apply to domestic C corporations. But, this can be modified in some cases by tax treaties with the country where the corporation is incorporated.
The definition of what constitutes "effectively connected income" is a bit arcane but basically involves income from either real estate or an active trade or business within the United States, as opposed to passive financial instrument income such as interest on a bond or promissory note, dividends, or capital gains on the sale of a security. There are many gray areas, which international tax lawyers make their livings cultivating and navigating.