FUG is a group that's against Zahard and the 10 Great Families. FUG wants to overthrow/kill Zahard so they can continue to climb the Tower. IIRC, Zahard has it so anyone can only climb up to Floor 134.
A regular is anyone is born in the Tower. An irreguar is anyone who comes from outside the Tower, like Rachel, Baam, Urek Mazino, Zahard, and the 10 Family Heads.
A Slayer is someone Regular or Irregular that is capable of killing Zahard or one of the 10 Family Heads. Zahard and 10 FH made a deal with the Administrators that no one from the Tower can kill them. They are for all intents and purposes, immortal and invincible. The Slayers are the loopholes.