The ban does not apply to green card holders, however it makes entry to the US for non-citizens more difficult due to additional background checks.
I hereby proclaim that the immigrant and nonimmigrant entry into the United States of aliens from countries referred to in section 217(a)(12) of the INA, 8 U.S.C. 1187(a)(12), would be detrimental to the interests of the United States, and I hereby suspend entry into the United States, as immigrants and nonimmigrants, of such persons for 90 days from the date of this order [ ... ].
Basically, the confusion is caused by whether the re-entry of US lawful permanent residents (or green card holders) are considered to be "immigrant entry".
Read up about this in @phoog's question on Law SE.
Initially, senior administration officials stated that green card holders will be included in the ban.
An article published by CNN states:
Senior officials at the Department of Homeland Security initially interpreted Trump's order to not apply to green card holders from the seven banned countries. Trump White House overruled that reading, however, meaning those green card holders were initially barred.
But a federal judge in New York temporarily blocked part of Trump's order late on Saturday night, ruling that citizens of the seven countries who hold valid visas and have already arrived in the United States cannot be removed from the United States.
Senior administration officials stated to CBS News Saturday:
Senior administration officials told CBS News Saturday that for permanent American residents -- those holding green cards -- from the listed countries, their readmittance to the U.S. will be done on a “case by case exemption process.”
There were reports that some green card holders were barred from reentering the US:
Up to 12 people were being held at New York's JFK Airport, two of them Iraqis who had worked for the US in Iraq and had been targeted for doing so.
[ ... ]
Green card holders were also being stopped and questioned for several hours. Officials also denied travellers with dual Canadian and Iranian citizenship from boarding planes in Canada that were headed the United States, she said.
After that, officials clarified that green card holders are allowed to enter the US:
The US government will not automatically allow green card holders who traveled to countries placed under a temporary travel ban back into the United States, two sources familiar with the matter told CNN.
Instead, after landing those travelers will have to apply for a waiver to the executive order that instituted the ban, the sources said. When they land, they will be taken into a secondary screening process in which they will be interviewed and their biometric and biographic information will be checked to see if there are any red flags. Then their eligibility to enter the country will be determined.
So, in conclusion, while US Permanent Residents are not banned from entering the US, the order makes the entry more difficult.
Basically, this CNN article sums up what we currently know and how the order is being carried out.
Republican Sens. John McCain of Arizona and Lindsey Graham of South Carolina released a joint statement Sunday saying: "Our government has a responsibility to defend our borders, but we must do so in a way that makes us safer and upholds all that is decent and exceptional about our nation.
"It is clear from the confusion at our airports across the nation that President Trump's executive order was not properly vetted. We are particularly concerned by reports that this order went into effect with little to no consultation with the Departments of State, Defense, Justice, and Homeland Security," they added.