What exactly is the President Pro-Tempore of the United States Senate?
For practical purposes, the president pro tempore is the highest ranking senator, as elected by the Senate. Traditionally, this is the senator who's served the longest among those in the party in power.
Officially, the primary duty of the president pro tempore, as set out in the Constitution (article I, section 3), is to preside over the Senate whenever the vice president isn't there to do so. In practice, both the vice president and the president pro tempore are often otherwise occupied, at which times the Senate appoints acting presidents pro tempore to do the job.
The president pro tempore of the Senate is third in the line of succession for the presidency, after the vice president and the speaker of the House.