According the the Constitution the President is the Commander-in-Chief. The congress appropriates monied and declares war. For an interesting read on this subject I'll post the following letter:
Dear William: Washington, Feb. 15. 1848[emphasis in original]
Your letter of the 29th. Jany. was received last night. Being exclusively a constitutional argument, I wish to submit some reflections upon it in the same spirit of kindness that I know actuates you. Let me first state what I understand to be your position. It is, that if it shall become necessary, to repel invasion, the President may, without violation of the Constitution, cross the line, and invade the teritory [sic] of another country; and that whether such necessity exists in any given case, the President is to be the sole judge.
Before going further, consider well whether this is, or is not your position. If it is, it is a position that neither the President himself, nor any friend of his, so far as I know, has ever taken. Their only positions are first, that the soil was ours where hostilities commenced, and second, that whether it was rightfully ours or not, Congress had annexed it, and the President, for that reason was bound to defend it, both of which are as clearly proved to be false in fact, as you can prove that your house is not mine. That soil was not ours; and Congress did not annex or attempt to annex it. But to return to your position: Allow the President to invade a neighboring nation, whenever he shall deem it necessary to repel an invasion, and you allow him to do so, whenever he may choose to say he deems it necessary for such purpose---and you allow him to make war at pleasure. Study to see if you can fix any limit to his power in this respect, after you have given him so much as you propose. If, to-day, he should choose to say he thinks it necessary to invade Canada, to prevent the British from invading us, how could you stop him? You may say to him, ``I see no probability of the British invading us'' but he will say to you ``be silent; I see it, if you dont.''
The provision of the Constitution giving the war-making power to Congress, was dictated, as I understand it, by the following reasons. Kings had always been involving and impoverishing their people in wars, pretending generally, if not always, that the good
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of the people was the object. This, our Convention understood to be the most oppressive of all Kingly oppressions; and they resolved to so frame the Constitution that no one man should hold the power of bringing this oppression upon us. But your view destroys the whole matter, and places our President where kings have always stood. Write soon again. Yours truly, A. LINCOLN
Abraham Lincoln, "Letter to William Herndon", The Collected Works Of Abraham Lincoln, Roy P. Basler, ed., Vol. 1, pp. 451-452 - Online here. FYI, Mr. Herndon was a Whig, Abraham Lincoln's jr. law partner.