For a given state, the senator with the most seniority (longest time served) in the Senate representing that state is the Senior Senator of that state. Overall seniority in the Senate is important for choosing committee membership, chairs, and office space. The party with the majority of seats in the Senate selects committee and subcommittee chairs, so generally the chairs are from that party. There are no exceptions to that in the current Senate. In the case of a tie vote, the VP can cast a vote to break that tie. I believe that is the only matter in which a VP's vote counts.
The term "class" refers to the Senator's class. That is, when is that Senate position up for re-election.
John