Senator Jefferson was a philandering politician and murder suspect that the Lone Gunmen investigated.
The four Lone Gunmen were trying to expose the Senator's affair with staffer Barbara Bonabo during a televised interview but their plan went awry and they went to jail. When they were bailed out, information had been left for them that led them to the location of the love child of Senator Jefferson and Barbara Bonabo. They believed that the Senator may have killed Bonabo and took the baby to protect the child.
As their investigation continued, it was revealed that her death had been an accident and that his campaign team, Brenda Sally and Jock Busfield, had covered it all up to ensure Senator's Jefferson's political victory.
When the truth came to his attention, the Senator did the right thing and revealed the truth publicly to voters. He came to see the Lone Gunmen and retrieve his son, naming him too, and they all learned that he had won the election despite the secrets revealed.
Behind the Scenes
Senator Jefferson is an obvious reference to Bill Clinton. Southern accent (sounds like the actor is doing the right region even), able to play the saxophone, extremely charismatic, adulterous, and high electability despite some of the darker shades about his character all make Senator Jefferson very much like Bill Clinton. They got an actor who looked like him and could imitate his mannerisms.
It should be mentioned that Clinton's full name is William Jefferson Clinton. Jefferson being the last name of the fictional senator here and William Jefferson being the name given to the baby.