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.
How does it affect refugees?
The ban completely suspends the United States' Syrian refugee programme, which accepted 12,486 Syrians in 2016. It also gives preference to accepting Christian refugees from the Middle East over Muslim refugees. And it reduces the cap on the total number of refugees allowed to enter the US in 2017 from 110,000 to just 50,000.
Is the executive order legally binding?
Executive orders are legally binding and are recorded in the Federal Register, a daily record of all federal regulations, proposals, and public notices.
But they can be subjected to a legal review - and, according to the New York Times, the "Muslim ban" order is illegal.
According to the newspaper, the Immigration and Nationality Act of 1965 banned all discrimination against immigrants on the basis of national origin.
Mr Trump's travel ban, it says, appears to be violation of that act.
What happens next?
Opponents of the "extreme vetting" order say they will launch a legal challenge on two fronts. They are expected to argue that the blanket ban violates the fifth amendment right to due process.
They will also argue that the order's preferential treatment of Christians over Muslims violates the first amendment on freedom of religion.