The presiding officer of the U.S. Senate is the person charged with maintaining order, interpreting Senate rules, and recognizing members to speak. The presiding officer is a role as opposed to an actual office.
The President of the Senate, as defined in the U.S. Constitution, is the Vice President of the United States. In the vice president's absence, the President pro tempore of the Senate, again, as defined in the Constitution, is the person "in charge," when it comes to the Senate as a body.
Presiding over the Senate is a job that is generally rotated through the members of the majority party on an ongoing basis.
"Today, I had the privilege of presiding over the U.S. Senate for the first time," Warnock tweeted early Tuesday afternoon. "Excited to get to work in this chamber to help Georgians overcome this once-in-a-century pandemic, expand affordable health care, strengthen the dignity of work & more."
Sen. Warnock, who was elected as Georgia's junior senator, replacing Kelly Loeffler earlier this month, was elected to the seat vacated by the resignation of Johnny Isakson for the term ending in January 2023.