Without going into a long explanation of the Constitution, they can be understood as an exercise of authority from the Executive Branch. Executive Orders are available publicly and can be found on the White House webpage.
The news calls them Muslim bans, which is understandable due to comments made during the last election process. In function, they do no ban Muslims. One of the first immigrant visas I obtained after the travel ban came into effect was to that of a Muslim coming from overseas.
Over time, the language and explanation of the purpose of the travel ban has become more clear. It's a rather dense process that requires input and evaluation from Homeland Security, Department of State and the Intelligence Community. They look at a variety of things like the ease of falsifying a country's passport, ability to obtain criminal and background information on citizens of the country and information sharing between countries.
So where are we now? If you or your family member is a citizen of one of the listed countries, you should familiarize yourself with the restrictions. For many situations there are waivers and steps that you can take to make travel smoother. Thing change fast, but as of today the listed countries are:
Executive Order 13780 (September 24, 2017)
- Chad (later removed in compliance)
- North Korea
Presidential Proclamation (January 31, 2020)
- Burma (Myanmar)