This term is used by the military forces of the United States of America to refer collectively to the current top layers of the operational and administrative command chains. Since the U.S. system places the ultimate control of the military underneath a civilian executive, this typically means the President and the Secretary of Defense, respectively.
Both are Executive Branch, and both are civilian. At any given moment, unless specifically delegated otherwise, the ultimate command authority over any U.S. military forces rests with those two offices. The President (POTUS in SecuritySpeak) sits atop the OPCON chain, with command over all military operations. The Secretary of Defense (SECDEF) and his office set policy for the U.S. Military - policy involving purchasing, doctrine, recruiting, organization and all the thousand and one associated areas in which a directed organization needs to explain to itself how to operate. These policies are typically closely tied to the goals of POTUS - SECDEF, after all, is a POTUS appointee.
The National Command Authority, or NCA for short, is most often referred to in any document or report which determines emergency decision-making. In the event of national-level security events (nuclear detonations, natural disasters large enough to require military response, other nations' military actions, problems with U.S. military forces, etc.) it is critical that the NCA be kept informed and that their authority not be undercut - or their orders countermanded by other levels of the chain of command.
The National Command Authority can devolve onto other persons or offices in particular instances. The continuity of government amendment determines how the offices of the Federal Government will shuffle in the event decision makers are either incapacitated or perish; however, in time-critical circumstances, NCA may devolve onto lower offices in the event that the current NCA is out of communication or otherwise unable to exercise command. The 'nuclear football' is perhaps the most visible symbol of National Command Authority - and hence its name. The 'football carrier' is the operational NCA.