Disclaimer: I am not an expert. I am an old fart that has been around the block a time or two. I am known to be wrong from time to time and my views are continually evolving.
The war on terror is a declared war. On September 20th, 2001, President Bush addressed both houses of the Congress declaring a War on Terror. Congress has continued to budget the military aspects of this war. Unlike other formal declarations of war, this war was not declared against a specific country but a network that is worldwide.
Like it or not, the President has the legal authority to target terrorist abroad in countries that have requested our assistance and even in countries that do not formally ask for assistance. As in other wars, American citizens have enlisted on the side of the enemy. In traditional wars they become military targets. Even though there is no formal battlefield, they become legitimate military targets in this war as they have in wars in the past.
Unlike other formal declarations of war, there is no formal battlefield. Also today drones are used that enable the targeting of specific targets limiting the amount of civilian collateral damage. Our enemy does not concern itself with limiting civilian collateral damage and targets areas where large numbers of civilians gather.
Terrorists exist because of real and perceived injustices. I would prefer that our war on terror tackle the real causes of terrorism. To often America is seen as the enemy because we back regimes that oppress their own people. I had hoped that President Obama would tackle this aspect of the war on terror. In some cases he has, but there are still to many oppressive regimes being supported by the United States. Our support of Israel's occupation of Palestine, our support of the repression of people by the governments of Bahrain, Saudi Arabia and other countries around the world are examples.