Clause 1. The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States; he may require the Opinion, in writing, of the principal Officer in each of the executive Departments, upon any Subject relating to the Duties of their respective Offices, and he shall have Power to grant Reprieves and Pardons for Offences against the United States, except in Cases of Impeachment.
Here the Framers spell out several of the president's more important powers. First and foremost, he is commander-in-chief of the military. Second, he is the boss of the heads of all the civilian departments of government; the bit here about requiring their written opinions provides the constitutional basis for the cabinet. And third, he has the power to pardon individuals convicted of crimes.
Clause 2. He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
More presidential powers: The president has the power to negotiate treaties with foreign governments, although a two-thirds vote of the Senate is required for ratification. He also has the power to nominate all appointed officials of the government, including both officers of the executive branch and judges of the judicial branch, although he needs to receive the "advice and consent" of the Senate in doing so. (In modern practice, that means a majority-vote approval of his nominees.)
Clause 3. The President shall have Power to fill up all Vacancies that may happen during the Recess of the Senate, by granting Commissions which shall expire at the End of their next Session.
The president can make "recess appointments"—appointing individuals to government positions without Senate approval—any time the Senate is out of session. Recess appointments are only temporary, however; they cannot last longer than the next session of Congress.