What type of power is the power to make treaties with foreign countries?

The power to make treaties with foreign countries is an expressed power that that Constitution gives to the President.

Article 2, Section 2 of the Constitution expresses the President's power in this way: "The President ... shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur." So, the President has the expressed power, vested in him by the Constitution, to make treaties with foreign countries, but then the Senate has the power to approve (ratify) those treaties.