The Senate holds a trial to assess the House’s charges — aimed at deciding whether to remove an impeached president from office.
In this trial, the House of Representatives acts as a prosecutor, and chooses certain “impeachment managers” to argue their case in the Senate. Then, the president’s lawyers are the defense team — the president does not have to appear in person and historically has not. The chief justice of the Supreme Court presides and is responsible for making procedural rulings during the trial — but the Senate can vote to overrule his decisions.
Now, though this is referred to as a trial, it is, again, a political and not legal process, so it doesn’t have to follow the ordinary rules and practices of a criminal trial. Again, it’s up the Senate to decide how to structure it — for instance, they can call witnesses to give live testimony (as they did for Andrew Johnson), or decide not to (as they did for Bill Clinton).
At the end, though, this trial ends in a vote on each article of impeachment — to either convict or acquit the president. A vote to convict on even one article will remove the president from office.
Senate Majority Leader Mitch McConnell showed no interest before the trial in subpoenaing witnesses who did not testify before the House. Ultimately, the Senate voted not to call any witnesses.