Jane Foster was travelling in a van with her colleague Erik Selvig and her assistant Darcy Lewis, when a mysterious weather event and the subject of Foster's research unexpectedly resulted in a powerful vortex before them. As they struggled to drive through zero-visibility conditions, the van collided with a man who had stumbled out of the dust and into their path. The man, who only announced himself as Thor, was injured again when Darcy tased him, prompting them to take him to hospital.
The following day, Darcy made a surprise discovery when she noticed the figure of a man in one of the photographic images captured from the atmospheric event. While Jane realized that the newcomer might be able to shed some light on the event, Selvig was more skeptical. So began a lengthy disagreement between the two, with Foster theorizing about the possibility of an Einstein-Rosen Bridge, while Selvig insisted that everything Thor had been saying about gods and other worlds was nothing more than delusional talk based on old myths.
Despite Selvig's concerns, Jane's interest in Thor pushed her to find out more, and the trio set off for the hospital to find the stranger. Once they were reunited, the trio and Thor went to a local diner where Jane pressed him for more information, trying to make sense of what he was telling them. It was there that Thor overheard news of his missing hammer, Mjölnir, and set off on his own to retrieve it. Returning home, Jane and her colleagues found S.H.I.E.L.D agents confiscating their equipment.
After commiserating with her colleagues over the loss of their property and telling Darcy to stop going on about her lost IPod, Jane chanced upon Thor again and volunteered to assist him with reacquiring his hammer Mjölnir. This goal proved unsuccessful when Thor, after stealing into the temporary S.H.I.E.L.D. compound that had sprung up around the artifact, found himself unable to lift it and allowed himself to be caught.
Jane enlisted Erik’s help in getting Thor released, which led to a series of events that bonded the expanded group more closely, with Selvig befriending Thor while Foster started to view him as a romantic interest. Their time together was cut short when first, several of Thor's friends from Asgard arrived to take him home, then the Destroyer (a very large battle automaton robot) arrived to stop him from returning to Asgard.
Thor ultimately beat the Destroyer. His willingness to sacrifice himself in the protection of those around him proved his worthiness of Mjölnir, which he regained along with his full power as the God of Thunder. Jane finally saw what he really was for the first time, but with the return of Thor's power came the return of his responsibility to his realm and his people. She was forced to say goodbye as Thor returned to his home.
Thor: The Dark World
To be added
While she does not appear in the film, it's revealed that she and Thor broke up.
Avengers: Infinity War
Though she doesn't appear in the film, it was later revealed that she was one of countless affected by the decimation, and was killed by Thanos.
When the Avengers travel back in time to collect the Infinity Stones, in order to undo the Decimation, Thor and Rocket Raccoon travel to Asgard in 2013 to retrieve the Reality Stone from Jane's body. They are successful in getting the stone out of her body before they travel back to the present. After undoing the Decimation and defeating Thanos, Captain America travels through time to return all of the Infinity Stones, allowing Jane to reabsorb the Reality Stone. Jane in the present time was revived by Hulk, but she doesn't appear at Tony Stark's funeral.
Thor: Love and Thunder
Jane Foster is a strong and determined woman, with an obstinate will and an independent spirit. She firmly believes that scientists have been invested with the mission of searching the truth and loyally follows this idealism.
She possesses a deep knowledge for her study field, astrophysics, and of her peers. She came the closest to discovering the existence of Asgard by studying the wormhole-like traces left by the use of the Bifrost Bridge.
In the comics, Jane Foster starts as a nurse, not an astrophysicist, but the Studios wanted to update the character. She is enamored with Donald Blake, having broken up with Thor decades ago in the comics.
Chris Hemsworth's wife Elsa Pataky was Portman's stand-in for the post-credits scene in Thor: The Dark World.