The President doesn't have the authority to declare war against another country, as that authority is given to Congress.
Today, however, the President does have the tools to launch different military actions within reason and for a certain amount of time, but must get the approval of Congress if the conflict ends up lasting longer than 60-90 days.
Throughout history in the United States, the powers of the President in regards to the use of the military and its faculties to carry on military operations have been constantly contested and still continue to be.
The War Powers Resolution of 1973 was set to address specifically the powers of the President, but there have been questions on whether it's in conflict with the Constitution or if it has also been violated by Presidents since then. We could say the adjustment of the War Powers Resolution to the Constitution is still an on-going process with a lot of merky areas surrounding it.
A good way to examplify this would be to look at the case of a president using his power regarding the military while respecting the authority of Congress, like President Theodore Roosevelt did regarding the Great White Fleet and it's world tour to display United States' naval power.
Congress contested that President Roosevelt didn't have the authority to do so for an extended period of time. Roosevelt didn't argue with them, instead just stated he had the power to send the navy out for a specific distance and a specific amount of time, and that it would be up to Congress to bring the fleet back home. Congress ended up approving the funding to do so.
Hope this helps out!
