At FBI headquarters, spokesman Paul Bresson declined to comment on the information that the affair had been discovered in the course of an investigation by the bureau.
Holly Petraeus is known for her work helping military families. She joined the new Consumer Financial Protection Bureau to set up an office dedicated to helping service members with financial issues.
Though Obama made no direct mention of Petraeus' reason for resigning, he offered his thoughts and prayers to the general and his wife, saying that Holly Petraeus had "done so much to help military families through her own work. I wish them the very best at this difficult time."
Petraeus, who became CIA director in September 2011, was known as a shrewd thinker and hard-charging competitor. Broadwell recently wrote a piece in Newsweek about his management style.
The article listed Petraeus' "rules for living." No. 5 was: "We all make mistakes. The key is to recognize them, to learn from them, and to take off the rear view mirrors — drive on and avoid making them again."
Petraeus told his CIA employees that he treasured his work with them "and I will always regret the circumstances that brought that work with you to an end."
The director of national intelligence, James Clapper, said Petraeus' departure represented "the loss of one of our nation's most respected public servants. From his long, illustrious Army career to his leadership at the helm of CIA, Dave has redefined what it means to serve and sacrifice for one's country."
Other CIA directors have resigned under unflattering circumstances.
CIA Director Jim Woolsey left over the discovery of a KGB mole and director John Deutch left after the revelation that he had kept classified information on his home computer.
Before Obama brought Petraeus to the CIA, the general was credited with salvaging the U.S. war in Iraq.
"His inspirational leadership and his genius were directly responsible — after years of failure — for the success of the surge in Iraq," Sen. John McCain, R-Ariz., said Friday.
President George W. Bush sent Petraeus to Iraq in February 2007, at the peak of sectarian violence, to turn things around as head of U.S. forces. He oversaw an influx of 30,000 U.S. troops and moved troops out of big bases so they could work more closely with Iraqi forces scattered throughout Baghdad.
Petraeus' success was credited with paving the way for the eventual U.S. withdrawal.
After Iraq, Bush made Petraeus commander of U.S. Central Command, overseeing all U.S. military operations in the greater Middle East, including Afghanistan and Pakistan.
When the top U.S. commander in Afghanistan, Gen. Stanley McChrystal, was relieved of duty in June 2010 for comments in a magazine story, Obama asked Petraeus to take over in Kabul and the general quickly agreed.
In the months that followed, Petraeus helped lead the push to add more U.S. troops to that war and dramatically boost the effort to train Afghan soldiers and police.
House Homeland Security Chairman Peter King, R-N.Y., said he regretted Petraeus' resignation, calling him "one of America's most outstanding and distinguished military leaders and a true American patriot."
Senate Intelligence Committee Chairman Dianne Feinstein also regretted the resignation but gave Morell high marks, too.
Morell had served as deputy director since May 2010, after holding a number of top roles, including director for the agency's analytical arm, which helps feed intelligence into the president's daily brief. He also worked as an aide to former CIA director George Tenet.
"I wish President Obama had not accepted this resignation," Feinstein said of Petraeus, "but I understand and respect the decision."
Associated Press writers Wendy Benjaminson, Kimberly Dozier, Ken Thomas, Donna Cassata, Eileen Sullivan, Pete Yost, Anne Flaherty and Julie Pace contributed to this report.