Most sanctions come from Congress (here's a list of pending legislation), however the President can, under executive authority, order some limited sanctions that would make life painful. As they are executive actions, however, a subsequent administration can undo them the same way.
The first step is to notify the Secretary of the Treasury. At present, that's Steve Mnuchin. Here's Trump's latest sanction request
President Donald Trump said Wednesday he ordered the Treasury Department to “substantially increase” sanctions on Iran.
Treasury's Office of Foreign Assets Control does the actual work of implementing sanctions. Here's their press release about Iran sanctions
Today’s action targets the CBI for its financial support to the IRGC-QF and Hizballah. In May 2018, OFAC designated the CBI’s then-Governor Valiollah Seif, and the Assistant Director of the International Department Ali Tarzali, for facilitating financial transfers for the IRGC-QF and Hizballah. Also, in November 2018 and as part of Treasury’s disruption of an international oil-for-terror network, OFAC designated the CBI’s International Department Director Rasul Sajjad, and the CBI’s International Department Director, Hossein Yaghoobi, for conducting financial transactions for the IRGC-QF.
As a result of today’s action, all property and interests in property of these entities that are in the United States or in the possession or control of U.S. persons must be blocked and reported to OFAC. OFAC’s regulations generally prohibit all dealings by U.S. persons or within (or transiting) the United States that involve any property or interests in property of blocked or designated persons.
In addition, persons that engage in certain transactions with the entities designated today may themselves be exposed to designation. Furthermore, any foreign financial institution that knowingly facilitates a significant financial transaction or provides significant financial services for entities designated in connection with Iran’s support for international terrorism or any Iranian person on OFAC’s List of Specially Designated Nationals and Blocked Persons could be subject to U.S. correspondent account or payable-through account sanctions.
The way this generally works is that OFAC notifies the Federal Reserve (central US bank) that certain people are not to have access to the US banking system. In addition, any assets they have here may be frozen or seized. They also notify the public such transactions are prohibited. If you violate the sanctions, you face penalties. You'll note some foreign banks are listed in there. They have to pay up, or risk losing access to the US banking system as well.