The Significant Mistake: As Junior Representative in the Galactic Senate, it was the gormless Gungan who granted Senator Palpatine emergency powers to use his clone army to fight Count Dooku's Separatists.  Worse, it's just about his only scene in Attack of the Clones. 
 
Impact On The Saga: Jar Jar's vote made the Clone Wars possible.  Y'know; two TV series, one spin-off film... Oh, and the small matter of Palpatine getting one step closer to becoming Emperor.
 
If The Force Had Been Strong With Him: It's hard to criticise Jar Jar for this, given that he shouldn't have been placed within 100 light years of anything approaching a position of responsibility.  But like so many others, he apparently can't tell from the sneering voice and malevolent stare that Palpatine's a bad'un.