That is... fuzzy to say the least. The constitution specifies that congress has the ability to "declare war", but that the president is head of the military (and the president does need the authority to deploy forces on short notice to handle emergencies, outside of formal wars.)
The line between a "war" and a simple military deployment is not always clear cut and hasn't really been tested in the courts.
The US hasn't formally declared war on anyone since 1942.