What is an executive order?
An executive order is an official statement from the president which tells government agencies how to use their resources. In the case of Mr Trump's "Muslim ban", the order bars citizens of seven Muslim-majority countries from entering the US for a period of 90 days.
It also suspends the United States' refugee system for a period of 120 days. Mr Trump says his "extreme vetting" system will help "keep radical Islamic terrorists out of the US".
Who does it affect?
The order itself does not name the countries whose citizens are banned from entering the US. Instead, it refers to a statute which applies to seven Muslim-majority nations. They are Syria, Iran, Sudan, Libya, Somalia, Yemen and Iraq, with dual nationals included in the ban.
Certain visa categories, such as those for diplomats, are exempt. There have also been reports of legal US residents, known as green card holders, being turned away from US-bound flights.
US homeland security secretary John Kelly said in a statement that people from the seven countries who hold so-called green cards as lawful permanent US residents would not be blocked from returning to the United States from overseas, as some had been following the directive.
Green cards are not mentioned explicitly in the executive order. Oddly, the ban does not apply to the nationalities of those who carried out the 9/11 attacks, such as Saudi Arabia, the United Arab Emirates and Egypt.