Canonically it was Bail Antilles of Alderaan who ran against Palpatine in the election for Chancellor, not Bail Organa. However there has been some confusion about the two characters, exacerbated by the fact that both appeared in the original shooting script for The Phantom Menace, but Bail Organa was cut from the movie and the actor originally meant to play him (Adrian Dunbar) never appeared on screen. The role of Bail Organa was then recast for Attack of the Clones, giving Jimmy Smits the role. Adrian Dunbar went on to represent Bail Antilles in various Star Wars Expanded Universe products, which confused some fans familiar with his original casting into thinking that the characters of Antilles and Organa were the same person.
We're doing our bit to add to the confusion.
Palpatine: M'Lady, this is Bail Organa, Senator for Alderaan.
Padmé: You'll vote for my army, right?
Bail Organa: Are you lookin' at me?
Padmé: No need to get all rebellious. Your planet will be spared if I have your vote.
Bail Organa: I don't want no trouble.
Palpatine: Ahem. Senator Organa was my worthy rival for the post of Chancellor in the election ten years ago.
Bail Organa: I coulda been the Chancellor. I coulda been somebody, instead of a Senator, which is what I am, let's face it.
Padmé: Senator's not bad, although you're probably not as powerful as me.
Bail Organa: You're tearin' me apart!
Padmé: Don't tempt me. Don't think I haven't noticed your goatee.