The Army created the first antecedent of the USAF in 1907, which through a succession of changes of organization, titles, and missions advanced toward eventual separation 40 years later. The USAF became a separate military service on September 18, 1947, with the implementation of the National Security Act of 1947. The Act created the United States Department of Defense, which was composed of three subordinate departments, namely the Department of the Army, the Department of the Navy and a newly-created Department of the Air Force. Prior to 1947, the responsibility for military aviation was shared between the Army (for land-based operations), the Navy (for sea-based operations from aircraft carriers and amphibious aircraft), and the Marine Corps (for close air support of infantry operations).
The predecessor organizations of today's USAF are:

Aeronautical Division, U.S. Signal Corps (August 1, 1907 to July 18, 1914) 
Aviation Section, U.S. Signal Corps (July 18, 1914 to May 20, 1918) 
Division of Military Aeronautics (May 20, 1918 to May 24, 1918) 
U.S. Army Air Service (May 24, 1918 to July 2, 1926) 
U.S. Army Air Corps (July 2, 1926 to June 20, 1941) and 
U.S. Army Air Forces (June 20, 1941 to September 17, 1947)