Well I've been curious whether SSB is considered a Platformer Game or a Fighting Game.

Like the Campaign is more of a platformer but when playing the insanely addictive Multiplayer, it's more of a fighting Game.

What is Super Smash Bros' genre?

It's a fighting game, with platforming element's. But it is first and foremost a fighter.

Well the objective is to defeat your opponent by beating them up, not by jumping on them.

fighting, the platforming is an extra

It's a Fighting game.

It's neither. It's not similar enough to traditional fighters (Street Fighter, Soul Caliber, ect.) to be in the same genre, but it certainly isn't a platformer. Instead, it created it's own genre, the Brawler.

*coughs*

Yeah, it's a fighter with platforming elements.

yeah, pretty much what koyotecat said. Fighter with platforming elements

Definitely a fighting game.. the single player is a platformer.

It's fighting. The scope of the fighting genre may have been narrowed somewhat due to the dominance of games like Streetfighter and then Tekken and Soul Caliber but it's still fighter. I mean even the 'platforming' side of the 1 player game is still interspersed with plenty of fighting.