By Roger H. Goun [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons
After impeachment passes the House, it goes to the Senate where a trial is held. Before the trial can start the Senate must make a set of rules for the trial. The case itself is a very complicated process, the House leadership would appoint a team that would act as the prosecutors for the case of impeachment and the President would recruit a team of lawyers to act as his defense.
The case would move forward as if it were a regular criminal trial, the judge would be the Chief Justice of the Supreme Court (currently John Roberts) and the jury would consist of the members of the Senate. Eventually, a verdict would be reached and the President would either be impeached or the President would remain in the office depending on the verdict.
Source: TTN staff