The man suspected of trying to bomb New York City's main bus terminal is 27-year-old Akayed Ullah, according to New York City Police Commissioner James O'Neill.
He was wearing an "improvised, low-tech, explosive device attached to his body", which he detonated intentionally, Mr O'Neill said.
The bomber suffered burns and other wounds and was taken to Bellevue Hospital in Manhattan.
Mr Ullah came from Chittagong, Bangladesh, and entered the US with his parents and siblings in 2011 on an immigrant visa, according to CBS News.
White House spokeswoman Sarah Sanders said Mr Ullah had entered the US on an F43 visa. This means he was the child of someone with an F41 visa, which is available to people who are the "brother or sister of a US citizen at least 21 years old".
Mr Trump has proposed eliminating so-called chain migration, which is when US immigrants legally bring family members into the country.
Mr Ullah was inspired by the Islamic State group but had no direct contact with it, law enforcement officials in the US told the Associated Press news agency.
He said he had been motivated by US air strikes on IS targets in Syria and elsewhere, the New York Times reports.
He told police investigators he had been inspired by Christmas terror attacks in Europe and selected the Port Authority bus terminal after seeing a number of festive posters on the subway walls.
Mr Ullah was a permanent US resident, living in Brooklyn, New York City.
Police in Bangladesh say he last visited that country on 8 September.
He lived in the same apartment building as his brother whom he recently worked with at an electrical company close to Port Authority, police said.
The New York City Taxi and Limousine Commission told CNN that Mr Ullah had held a taxi driver's licence from March 2012 to March 2015.
However, he did not drive a New York yellow taxi or work for Uber.
The Inspector General of Police in Bangladesh, AKM Shahidul Haque, said Mr Ullah had had no criminal record in the country.
Bangladesh police are continuing to investigate Mr Ullah's background, police said.
Mr Ullah has, up to now, also held a clean record in the US, with just traffic violations cited by the police.