Continental United States

The continental United States refers (except sometimes in U.S. federal law and regulations) to the largest part of the U.S that is delimited by a continuous border, also referred to as the conterminous states, the coterminous states, or the contiguous states, and

in various specialized contexts, as "CONUS",
in Hawaii, as "the mainland" or "the continent",
in Alaska, as either "the lower 48", or sometimes "outside", and
also occasionally within the conterminous U.S., as "the lower 48".

Specifically, this includes 48 states and the federal capital of the U.S., the District of Columbia; it unambiguously excludes the remaining two states: it is clear that neither Hawaii (an archipelago in the North Pacific Ocean), nor Alaska (being separated from the "lower 48" by Canada), is either contiguous or conterminous with any other U.S. state.

Each of these terms has some shortcoming of illogic, ambiguity, or excessive or deficient formality. In particular:

no collection of states includes the District of Columbia;
both "conterminous" and "coterminous" are rare, somewhat technical words;
"contiguous" has a more usual sense, narrower than "conterminous", that applies to areas that touch each other but not to areas that can be reached from each other only via an intervening chain of touching areas;
while Hawaii is not part of any continent, Alaska is clearly, like the contiguous states, part of North America, and excluding it from the "continental U.S." must be described as a misnomer.

Use in Federal Law

As the language of the Alaska Omnibus Act of 1959 makes apparent, the term was in use in U.S. federal law prior to then. It presumably dates from after the acquisition of Alaska in 1867, and probably from after the Spanish-American War brought the U.S. its first off-continent possessions in 1897. Whatever else these terms may be, "continental United States" is a term defined in various federal laws, in different ways in different time periods; it is also defined in different ways at the same time, depending on whether the context was the IRS or not, during at least a period that began with Alaska statehood.