By the Constitution it is Article 2, Section 4: "The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors."
... and Article 1, Section 3: "Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States: but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment and Punishment, according to Law."
By process (best I can tell) the House has to have a formal process of Impeachment against the President, then it moves to the Senate where the President is defended by his counsel against House counsel acting as the prosecution. The Chief Justice presides the case, the Senate is turned into a jury. As a jury, they must debate and reach a verdict and 2/3 vote is required for conviction. Then a second vote is determined for removal from office as a result of conviction.
Now this is gray area on the determination on what qualifies as a "crime." Also, valid impeachment proceedings have been started in the past that did not result in a President being removed.