The Legislative branch: H.R, (House of Representative's) Senate.
The Supreme Court plays no role in impeachment trials. However, in the impeachment trial of the President of the United States, the Chief Justice of the United States serves as presiding officer of the Senate since it would be a conflict of interest to have the vice president presiding over a trial at which he would become President if the current President were to be found guilty.
The US Senate serves as the court, and the jury, for the trial that follows impeachment (accusal) by the House of Representatives.
He has no role unless he is the one who is impeached. In this case, he stands trial in the Senate and presumably would be called to testify, if he wants to. He would be able to mount a defense.
All federal impeachment trials are heard by the U.S. Senate, which serves as jury. A two-thirds vote is required for conviction. In the case of presidential impeachment trials, the Chief Justice of the U.S. Supreme Court presides.
The Chief Justice of the United States presides over a US Presidential impeachment trial; an "Impeachment Trial Committee" comprising Senators selected by the Presiding Officer of the Senate* is in charge of all other impeachment trials. This practice began in 1986 when the Senate amended its rules and procedures for impeachment trials.* The Presiding Officer is not a fixed position. The Vice-President (currently Joe Biden), who also serves as President of the Senate, could be the Presiding Officer; or, in his (or her) absence, the President pro tempore (currently Daniel Inouye, D-HI) could preside; or, anyone the President of the Senate or President pro tempore designates -- even a Junior Senator -- could serve in that capacity.
the vice president serves as president of the senate
The full Senate serves as the jury in any impeachment trial. Most impeachment proceedings have been conducted against federal court judges (below the Supreme Court), so the Senate streamlined the trial process to allow the majority of Senators to conduct the chamber's regular business while an "Impeachment Trial Committee" listens to testimony and acts as a panel of judges. At the end of the trial, the committee compiles a report which the full Senate reads before voting for conviction or acquittal.
This depends on whom you are talking about impeaching. For the impeachment of a Federal Official such as, but not only the President of the United States. In Federal cases, the House of Representatives impeaches the Official by a simple majority vote of the House. Impeachment does not mean the Official is removed from office. This simply serves as the indictment of the Official. After the impeachment, the Senate of the United States will conduct a trial of the charges against the accused. The Chief Justice of the Supreme Court presides over the trial.
The US does not have a prime minister. The President and Vice president are elected separately from the Congress, and the President appoints his cabinet secretaries (ministers) from both elected officials and private citizens. Those appointed to the executive branch must resign any Congressional or other governmental post. The Congressional leader of the House of Representatives is the Speaker of the House. The Vice President of the US serves as the President of the Senate, but his functions may be fulfilled by the President pro tempore (senior senator).
US Senate
There is always a congressional election held at the same time as the presidential election. However, since Congressmen serve for only a two-year term and the president serves for four years, there is a another Congressional election in the middle of the president's term. ( I am talking about the lower house of Congress. The senate elects one-third of its members every two years, so one-third of the senators are also elected at the same time as the President. )
The Chief Justice of the United States presides over the Senate trial, when the President has been impeached. If the president is convicted and removed from office, the vice-president would become President.