"The Senate shall have the sole Power to try all Impeachments. When sitting for that Purpose, they shall be on Oath or Affirmation. When the President of the United States is tried, the Chief Justice shall preside: And no Person shall be convicted without the Concurrence of two thirds of the Members present."
Comment: This clause vests to the Senate the powers and procedures regarding impeachment trials. The Senate has the "sole" power to try Impeachments for a couple of reasons. First, the Framers thought that the Senate would be smarter and more level-headed than the House of Representatives. Second, the Framers thought the Judiciary was too small to trust and that a "numerous court for the trial of impeachment" was more appropriate, as explained in The Federalist No. 65.
There are three requirements for impeachment trials. First, the Senators must be under Oath or Affirmation, which is to impress upon the Senator the importance of the occasion. Second, the Chief Justice must preside over the presidential impeachment trial, which avoids the potential conflict of interest of having the Vice President presiding over the Senate, as (the VP normally presides as provided in Article 1, Section 3, Clause 4). Third, to convict there must be a super-majority, which is designed to make sure the consensus would transcend party lines.
A couple of issues have arisen regarding the impeachment authority. One was the Chief Justice's authority to hand down unilateral rulings. In the first presidential impeachment trial Chief Justice Chase had many rulings challenged and two were overruled. However, during President Client impeachment trial, Chief Justice Rehnquist was neither challenged nor overruled for any of his rulings.
Another issues had to do with which procedures the Senate must utilize during trials. Under Article 1, Section 5, Clause 2, the Senate has the power to determine the Rules of its Proceedings. To that end, Thomas Jefferson, while Vice President, formulated the Senates impeachment trial procedures, which have subsequently been amended - but have largely remained intact ever since and were followed in President Clinton's impeachment trial.
Two U.S. Presidents have been impeached: Andrew Johnson and Bill Clinton. Both were acquitted at trial.
Primary Source. The Heritage Guide to the Constitution.