National Office for Combatting Terrorism
The National Office for Combatting Terrorism within the Executive Office of the President was introduced on September 21, 2001, as S. 1449 by Senators Bob Graham, Dianne Feinstein, Bayh, Barbara Mikulski, Dick Durbin, Bill Nelson, and Jay Rockefeller in the U.S. Senate and on October 10, 2001, as H.R. 3078 by Representative Hastings (of Florida) in the U.S. House of Representatives during the First Session of the 107th Congress.
The first Director was retired Army General Wayne A. Downing who "was tapped in early October  and charged with starting up the White House Office for Combatting Terrorism. ... White House spokesman Ari Fleischer announced on June 27, 2002, that Downing had quit."
Retired Air Force Gen. John A. Gordon was appointed by President George W. Bush as Downing's replacement "to the position, which coordinates military, diplomatic, law enforcement, intelligence, financial and strategic information activities meant to deter terrorism." Gordon had served as "the Department of Energy's undersecretary for nuclear security and administrator of the National Nuclear Security Administration." He had also served as "a top National Security Council adviser" under George H.W. Bush as a deputy director of the Central Intelligence Agency under Bill Clinton. 
In the meantime, the initial piece of legislation was followed on April 11, 2002, by DRAFT National Homeland Security and Combating Terrorism Act of 2002 which was sponsored by Senators Joseph I. Lieberman, D-Conn., Arlen Specter, R-Pa., and Bob Graham, D-Fla. 
- "The proposed  legislation calls for formation of a Department of National Homeland Security, at cabinet level, to plan, coordinate, and integrate U.S. Government activities relating to homeland security, including border security and emergency preparedness, and to act as a focal point regarding natural and manmade crises and emergency planning. The legislation, which will be introduced shortly, creates a White House Office of Combatting Terrorism to develop an overall assessment of terrorist threats, to craft and oversee a National Strategy to Combat Terrorism, and exercise budget certification authority over spending to combat terrorism."  Also see April 11, 2002, press release (cache file).
According to an April 7, 2004, Reuters' news report, the attrition among all levels of the Office for Combating Terrorism began shortly after the attacks and continued into this year. At least eight officials in the office -- which numbers a dozen people -- have left and been replaced since 9/11. Several of the officials were contacted by Reuters.
The office has been run by four different people since the attacks, and at least three have held the No. 2 slot.
"There has been excessively high turnover in the Office for Combating Terrorism," said Flynt Leverett, who served on the White House National Security Council for about a year until March 2003 and is now a fellow at the Brookings Institution think tank.
"If you take the (White House) counterterrorism and Middle East offices, you've got about a dozen people ... who came to this administration wanting to work on these important issues and left after a year or often less because they just don't think that this administration is dealing seriously with the issues that matter," he said.
Some also left because they felt President Bush had sidelined his counterterrorism experts and paid almost exclusive heed to the vice president, the defense secretary and other Cabinet members in planning the "war on terror," former counterterrorism officials said.
Roger Cressey, who served under Clarke in the White House counterterrorism office, said: "Dick accurately reflects the frustration of many in the counterterrorism community in getting the new administration to take the al Qaeda issue seriously."
Cressey left the office in November 2001, when he became chief of staff of the White House's cybersecurity office until September 2002
Office of Cyberspace Security
A news story compiled by wire services, found September 30, 2001, on the Office of Cyberspace Security web site, stated that Richard A. Clarke, a "holdover from the Clinton administration," was to "oversee Cybersecurity for President Bush as head of a new Office Of Cyberspace Security, while a retired U.S. Army general will coordinate anti-terror efforts with military and intelligence counterparts."
Clarke, who had "served as counter-terrorism chief at the White House for more than a decade" and had been "appointed by President Clinton as the first national coordinator for security, infrastructure protection and counterterrorism," at the time of his appointment headed the government's counterterrorism team. He was to "direct efforts to protect the nation's information infrastructure from attack," according to three administration officials.
"Retired Army Gen. Wayne Downing will coordinate intelligence and military resources in the anti-terror campaign," and both Clarke and Downing were to "serve Bush at the National Security Council, working alongside Tom Ridge, who was tapped by Bush to head a newly created, Cabinet-level Office of Homeland Security. While the officials said their precise roles have yet to be determined, two other White House aides said Clarke and Downing will work beneath Ridge in the anti-terror hierarchy.