In the majority opinion, Chief Justice John Roberts wrote that the ban was "squarely within the scope of Presidential authority under the INA," referring to the Immigration and Nationality Act.
Read the court's full opinion in the case here.
Read Justice Kennedy's full concurring opinion here.
In one of two dissenting opinions, Justice Sonia Sotomayo who was joined by Justice Ruth Bader Ginsburg, said the court's decision "fails to safeguard" the "principle of religious neutrality in the First Amendment. It leaves undisturbed a policy first advertised openly and unequivocally as a 'total and complete shutdown of Muslims entering the United States' because the policy now masquerades behind a facade of national-security concerns". Read her full dissent here.
The Travel Ban remains in effect, and travel of citizens or nationals of banned countries to the USA remains either fully banned (all visas) or partially banned (only some visas are not allowed, while others are still permitted).
The full list of the banned countries and the kinds of visas banned for each country is here.