In addition to the information below, you might want to read the travel advisory issued by NAFSA: Association of International Educators.
Ban on entry to the U.S.
The executive order bans entry for at least 90 days from January 27 for nationals from Iran, Iraq, Libya, Somalia, Sudan, Syria, and Yemen. If you are from one of these countries, you can stay in the U.S. provided you maintain your current immigration status.
Travel outside the U.S.
We strongly recommend that you not travel to any country outside of the U.S. at this time. If you do, you will not be able to reenter the country, at least until the ban on entry to the U.S. is lifted.
U.S. permanent residents
U.S. Customs and Border Protection agents have broad discretion and authority to question an individual regardless of country of origin or nationality. Although we have received word that U.S. citizens and legal permanent residents should not be affected by this order (read the DHS statement), we recommend against travel unless necessary.
Naturalized citizens and dual citizens
DHS has indicated that you should be able to travel without issue if the U.S. visa is in your non-banned passport. However, we cannot guarantee that you will not have additional delays during transit, and thus we would remind you to proceed with caution.