A declaration of war is a formal declaration issued by a national government indicating that a state of war exists between that nation and another.
The United States Declares War on Germany I THE GREAT WAR Week 141 by The Great War
The document Declarations of War and Authorizations for the Use of Military Force: Historical Background and Legal Implications gives an extensive listing and summary of statutes which are automatically engaged upon the US declaring war.
The Articles of Confederation - I: Becoming the United States - Extra History by Extra Credits
For the United States, Article One, Section Eight of the Constitution says "Congress shall have power to...
However, that passage provides no specific format for what form legislation must have in order to be considered a "declaration of war" nor does the Constitution itself use this term.
In the courts, the United States Court of Appeals for the First Circuit, in Doe v. Bush, said: "[T]he text of the October Resolution itself spells out justifications for a war and frames itself as an 'authorization' of such a war."
The United States Court of Appeals for the First Circuit is a federal court with appellate jurisdiction over the district courts in the following districts:
in effect saying an authorization suffices for declaration and what some may view as a formal Congressional "Declaration of War" was not required by the Constitution.
This article will use the term "formal declaration of war" to mean Congressional legislation that uses the phrase "declaration of war" in the title.
Elsewhere, this article will use the terms "authorized by Congress", "funded by Congress" or "undeclared war" to describe other such conflicts.