Original series
Jessie received Ekans for chūgen (as a birthday present in the dub) before she met Ash. It is unknown who it was a gift from.
Arbok first appeared in Pokémon Emergency! as an Ekans. Throwing their Poké Balls down from their Meowth balloon, Jessie and James used both it and James's Koffing to quickly overtake the defenseless Viridian Pokémon Center. Koffing's job was to create a SmokeScreen while Ekans destroyed the computers. Chasing everyone into the back room, Ekans scared away the Pokémon that were sent out by Ash, which causes Misty to decide to battle Team Rocket. This is a failed effort, however, as both of them manage to catch up to Ash in no time at all. Just then, Nurse Joy's Pikachu come out and heal Ash's Pikachu with a Thunder Shock attack, while at the same time attacking the Team Rocket. Giving Ash's Pikachu enough power to stand up, Ash charged-up Pikachu using Misty's charred bike, and he let him have his own Thunder Shock. As the group are being electrocuted, Koffing releases a stream of Smog from one of the craters on its body, and the electricity climbs up the toxic gas, causing the entire Pokémon Center to explode, and sending Team Rocket blasting off.
It was used constantly by Jessie and eventually evolved into an Arbok in the episode Dig Those Diglett! along with James's Koffing after Jessie and James desperately wanted their Pokémon to evolve.
Although Arbok lost nearly every battle, as is traditional for Pokémon owned by Team Rocket, it has occasionally been shown to have extraordinary power and endurance to match its massive size. For example, in The Misty Mermaid, Arbok was able to single handedly battle Horsea, Starmie, Squirtle and Seaking underwater (where water Pokémon have the advantage) without the help of Weezing, enduring a huge amount of offense and eventually cornering all its opponents for the victory before Seel evolved into Dewgong and defeated it with Ice Beam.
Jessie used Arbok as her main battling Pokémon throughout most of the series up until Advanced Generation, a role which was taken over by Seviper. Jessie also used it during the Princess Festival.
Advanced Generation series
In A Ruin With A View, Jessie sent out Arbok alongside James's Weezing to battle Team Magma grunts who approached them. Not long after being sent out, Team Magma's Houndoom used Roar to send Arbok and Weezing back into their Poké Balls.
In A Poached Ego!, Team Rocket found a poacher holding a herd of Ekans and Koffing captive in Hoenn. After freeing them, James and Jessie reluctantly ordered Arbok and Weezing to stay in the wild and guard their fellow Pokémon. The two Pokémon have not been seen since.
Jessie and Arbok shared a close bond, as demonstrated when Arbok refused to leave her at first and cried when it finally did.
Arbok and Weezing seem to be both very emotional. This was first shown in Island of the Giant Pokémon, where they got very sad for being separated from Jessie and James, crying desperately, with Pikachu trying to cheer them up. At the end of the episode, when every lost Pokémon got reunited with their partners, Ekans and Jessie, along with James and Koffing, were happily crying, while Ash was only smiling with Pikachu, which may mean that Jessie's bond with Arbok was stronger than Ash's bond with Pikachu. It was also in this episode that it explained to Meowth along with James's Koffing "Pokémon do bad things because master bad. Pokémon not bad." In Lights, Camera, Quacktion both Arbok and Weezing got depressed for being rejected as the main characters in the movie. It was only with Jessie's and James's help that they got over it. On their last appearance, in A Poached Ego!, they were reluctant in leaving Jessie and James, and when they did, they strongly cried.