Grandparents, aunts, uncles, nephews and nieces are not considered to be "bona fide" relations.
The rules apply to people in Iran, Libya, Syria, Somalia, Sudan and Yemen, as well as all refugees.
Moments before the ban began at 20:00 Washington time (00:00 GMT), it emerged that the state of Hawaii had asked a federal judge for clarification.
It has in the past accused the US government of violating the Supreme Court's instructions by improperly excluding people.
Earlier this week, the Supreme Court partially upheld the ban, lifting injunctions that had halted one of the president's key policies since February.
The court ruled that people seeking visas to travel to the US from the six restricted countries, and all refugees, would have to prove a "bona fide relationship" to someone in the country.
The Supreme Court is expected to make a final decision on the ban in October.
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Who can come in?
According to the new rules, for the next 90 days those from the six countries without a close relationship will not be able to enter the US.IN - a parent, spouse, child, son or daughter-in-law, or sibling, including step- or half-siblings.
OUT - grandparents, aunts, uncles, nieces, nephews, in-laws, extended family and grandchildren.
Also exempt from the new rules are those with business or educational ties to the US.
However, the guidelines specifically state that the relationship must be formal, documented and not formed for the purpose of evading the order.
Those who already hold valid visas are not affected. Dual nationals who travel on their passport from the unaffected country will also be allowed entry.
The court also approved a 120-day ban on refugees entering the US, allowing the government to bar entry to refugee claimants who cannot prove the same ties to an American individual or entity.
More on the travel ban
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