The Acting President of the United States exercises the office of President of the United States while the President is unable to do so, as determined by the Twenty-fifth Amendment to the United States Constitution.
Article II section 1, and Article I section 3, of the Constitution apply to the office of Acting President, but have been vigorously debated since John Tyler's days. The Twenty-fifth Amendment, ratified in 1967, clears up many of the issues which surrounded presidential succession and incapacity. Only three times since the passage of the 25th Amendment in 1967 has someone acted as President and in all cases the self-declared incapacity method was voluntarily used while the President was to be under anesthesia.
Article II, section 1 of the Constitution, provides for the function of the Vice President serving as acting President:
In case of the Removal of the President from office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected. (Emphasis added.)
Article I, section 3 of the Constitution, provides for the election of the officers of the Senate, and declares that the Vice President concurrently holds the office of President of the Senate. Provision is further made for when the Vice President is absent from presiding over the Senate, or when he shall exercise the office of President. But the Vice President does not vacate the office of Vice President, President of the Senate nor does he ascend to the office of President, but merely carries out the "Powers and Duties of the said office" [Article II, Section 1, clause 6]:
The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.