In the original film, before Jafar became a sorcerer, the staff was shown to have the ability to use hypnosis, as shown when Jafar used the staff to get the Sultan to hand over his mystic blue diamond ring, claiming he needs it to "find the princess a suitor", when he actually needs it to discover the identity of "the diamond in the rough."
In another scene, Jafar attempted to use the staff's power to have the Sultan allow him (Jafar) to marry Princess Jasmine. However, the fanfare caused by the arrival of Prince Ali (Aladdin) distracted the Sultan before he could finish his hypnotized response.
He later brainwashed the Sultan shortly after attempting to have Ali killed, trying to have him state that Jasmine must marry Jafar, and later after Ali revealed himself to have survived and exposed that Jafar had tried to have him killed, to claim that Ali was "absolutely lying". However, Ali, deducing how Jafar was able to persuade the Sultan, destroyed the staff, with Jafar exposed for his villainy as a result.
Later on, Jafar steals Aladdin's lamp, taking over his role as the Genie's master. He uses his second wish to become the most powerful sorcerer in the world. His staff was restored but it now has open fangs and was extremely powerful that used many skills including moving people and objects. During the final battle, the staff disappeared when Jafar become a giant snake, and then an all powerful genie.
In the sequel, Jafar is freed from his lamp, and after transforming into his human form, he retained his staff and it was by his side throughout the entire film. It's original purpose was also alluded to by Iago and Aladdin, where the former claimed he had been a victim of the hypnotic properties, and Aladdin also believed it after he had been saved by Iago and tried to tell the Sultan this was the case with them not believing him. It had its original look and shot laser bolts from the eyes. He was also shown holding it in his Genie form. It is unknown what happened to the staff after Jafar's death, but it could be assumed that it was destroyed along with or after his lamp.
In the Aladdin and Hercules crossover, the staff was given its largest role and once again had its original look. In the episode, Hades, Lord of the Underworld, decided to revive Jafar from death after his demise in The Return of Jafar. The staff was used to keep him alive, and if he ever lost it, he would become a lifeless spirit once more. This staff was extremely powerful; creating things like live scorpions and shot fireballs. This staff was once confiscated by Hades after Jafar, in hysteria, started causing destruction in the underworld, as a means to force him to cooperate.
However, the creations of the staff can only exist as long as the staff exists, otherwise the creatures will fade out of existence. Aladdin and Hercules destroyed it during their battle against Jafar and Hades, and with it, Jafar was killed once again and his soul was sucked into the River Styx for good.
In the spinoff show to Once Upon a Time, Jafar's staff was actually a woman named Amara who was trapped in the form of a serpent staff by Jafar. Amara was a powerful healer and sorceress who is feared amongst the residents of Agrabah.
In her earlier life, she was the mother of Cyrus, Taj, and Rafi. When she was badly burned upon some people that Cyrus beaten in a card game, Cyrus had to use the water from the Well of Wonders to heal her. The effects of the water work heal Amara as she learns what had happened. When Nyx arrives, she begs her to spare her sons as they are turned into genies with their lamps being scattered across Agrabah.
A young Jafar visits Amara at her home, begging for her to teach him magic to exact revenge on his father the Sultan for abusing and abandoning him. She insists he is not ready, though he insists that he is, and she reluctantly begins teaching him dark magic. As a man, Jafar is tricked by Amara into giving his friend Akil a poison to make him see how far he will need to go to become the sorcerer he intends to be. The two then begin a sexual relationship and begin a plan to obtain genie magic to become the most powerful sorcerers in the world. Jafar and Amara successfully obtain a genie bottle each, though Jafar learns of a third in Agrabah. He then tricks Amara by siphoning her magic, transforming her into a snake, and converting her into his serpent staff.
When Jafar manages to steal the Knave of Hearts' heart from Alice and Cyrus, he tries to blast them only to have Amara's staff form work against him, forcing Jafar to retreat with the Knave of Hearts' heart.
Amara was later freed from her staff form due to the hints given to Alice, Will, and Cyrus by the Jabberwocky. Amara confronted Jafar and they both ended up in a brief duel until Cyrus got wounded by a shard of glass. Jafar convinced Amara that they should use the powers of the genies like they planned to do in order to save Cyrus' life.
When it came to Cyrus taking Amara to the Well of Wonders to return the water in her back to Nyx, Jafar attacked them which resulted in Amara dissolving into water upon her death. Following the final battle with Jafar, Cyrus told Taj and Rafi that they were freed from the genie curse at the cost of their mother's life.
Why Jafar never attempts to hypnotize Jasmine with the Snake Staff can be interpreted like this. It's vaguely implied that the Snake Staff only works on weak-willed people (e.g. the Sultan) and the rebellious Princess Jasmine is too strong-minded. Even the Sultan does manage to snap out of the hypnosis on his own at least once. It also only seems to affect the victim while he's actually holding it in their face, making it impractical for long-term work.
After Genie grants Jafar's wish to become the most powerful sorcerer in the world, the Snake Staff's appearance changes slightly, with the mouth opening to reveal four fangs. It remains that way for the remainder of the film.