Under Article One, Section Eight of the Constitution,
The Congress shall have power...
To declare war
but does not say what follows from "declaring war", nor does it say what form such a declaration shall take. This rather long document analyzes the notion of "declaration of war". The primary significance of a "declaration of war" lies in international law, see the Hague Conventions of 1899 and 1907. Articles 1 and 2 say:
The Contracting Powers recognize that hostilities between themselves
must not commence without previous and explicit warning, in the form
either of a reasoned declaration of war or of an ultimatum with
conditional declaration of war
and
The existence of a state of war must be notified to the neutral Powers
without delay, and shall not take effect in regard to them until after
the receipt of a notification, which may, however, be given by
telegraph. Neutral Powers, nevertheless, cannot rely on the absence of
notification if it is clearly established that they were in fact aware
of the existence of a state of war
So if the US were to declare war on Canada, we would need to officially notify the government of Canada of this fact, which we could do because there is a country Canada whose government we can inform. This is not possible with "terrorism", which is not a nation. There were official declarations for the War of 1812, against Mexico, Spain, and WW I, WW II (11 declarations). Since "the South" was not (from the perspective of "the North") a separate country, the United States did not declare war on itself to conduct the Civil War. In fact, since the congressional action authorizing military action against the Algerine Cruisers (the 2nd Barbary War) in 1815, the actions authorized by Congress have not used the word "war" (except in mentioning a specific law, the War Powers Act), except in the aforementioned declared wars. Instead, the resolution described what could be done:
it shall be lawful fully to equip, officer, man and employ such of the
armed vessels of the United States as may be judged requisite by the
President of the United States for protecting effectually the
commerce and seamen thereof on the Atlantic Ocean, the Mediterranean
and adjoining seas
"Declared war" is a fairly limited genre of use of force, but "being at war" in the broadest sense, is common and can involve concepts and things (terrorism, poverty, drugs).