What does the Senate do during the impeachment process?

It sits as a jury does in a court case, hears the evidence, and votes to determine the outcome.
The impeachment process is a 2-step process, much like a criminal trail. The House of Representatives acts as a grand jury. It hears testimony, seeks evidence, and ultimately decides if there is enough evidence to impeach the president (a simple majority is needed) and proceed to a trial.
The Senate is essentially the trial jury, presided over by the Chief Justice of the U.S. Supreme Court. TH trial proceeds and, in the end, the senate votes whether or not to remove the president from office. A 2/3’s vote is needed to convict the president and strip him of his office.