Comic Relief Disaster
In Star Wars: The Phantom Menace, Qui-Gon Jinn encounters Jar-Jar shortly after arriving on the surface of Naboo. Qui-Gon saves the Gungan from being run down by a Trade Federation troop transport, and the Gungan attaches himself to the Jedi Knight in order to repay the "life debt" that he has incurred. While occasionally useful as a native guide and contact with the Gungan leadership, Jar-Jar is also a perpetual nuisance and generally incompetent. At one point, he is appointed "Bombad General" by Boss Nass, but he soon proves as inept in this role as any other.
The character seems to exist largely to provide comic relief in the form of physical humor.
In Star Wars: Attack of the Clones, Jar-Jar mercifully plays a far smaller role. He is part of Senator Amidala's entourage on Coruscant, and she inexplicably leaves him to speak for her when she returns to Naboo to hide from assassins. Supreme Chancellor Palpatine easily manipulates Jar-Jar into proposing the Senate motion that will grant him emergency powers to raise an army with which to fight the Separatists, thus making Jar-Jar instrumental in the downfall of the Republic.
Jar-Jar appears briefly in Star Wars: Revenge of the Sith as a mourner at Padmé Amidala's funeral.
Jar-Jar occasionally appears in Star Wars: The Clone Wars. The writers of the show use him sparingly, and when they do, they generally manage to make him slightly less annoying.
The ASVS Theory
A popular theory on ASVS around the time that TPM was released was that Jar-Jar was supposed to be a loveable character that the audience could identify with and be sympathetic to, making it a shock for him to be eventually murdered by Anakin when he turned to the Dark Side. Unfortunately, the unpopular reaction to Jar-Jar in TPM meant that the audience would likely cheer at his death. He was relegated to background shots and single lines in AOTC and ROTS, removing him from assassination.
The Sith Lord Theory
Another theory holds that Jar-Jar Binks is actually an extremely powerful Sith Lord. In TPM, he performs feats of physical strength and agility that would normally be expected of Jedi or Sith, and in the Battle of Naboo he conveniently destroys everything that threatens him in his apparent clumsiness. The theory holds that Jar-Jar's sinister nature would be revealed in Episode 2 or 3, but the rejection of Jar-Jar by the fan base caused Lucas to reconsider his plans and introduce Count Dooku as the villain of Episode 2.