The Phoenix Force may currently be possessing several members of the Avengers, but the Marvel character most famous for being its host will always be Jean Grey. She was the character through which the Phoenix was introduced and, to many, Jean and the Phoenix are one and the same. However, that's not actually the case, and when Jean made a point to separate herself from the Phoenix when she returned from the dead in Matthew Rosenberg, Leinil Francis Yu, Joe Bennett, Carlos Pacheco and Ramon Rosanas' 2017 event miniseries Phoenix Resurrection: The Return of Jean Grey, although recent events suggest they still share a connection of some kind.
Created by Chris Claremont and Dave Cockrum the Phoenix Force first appeared in Uncanny X-Men #101 in 1976. After a mission to space went sideways the X-Men's ship made a crash landing. As her friends were searching for her, a version Jean suddenly emerged from the water in a new costume and announced that she was fire and life incarnate before collapsing. It was revealed that the Phoenix Force had sensed Jean's mental cries for help during the crash and bonded with her.
Later retcons would complicate this explanation by saying that Jean's real body had actually been stored in a cocoon and that the one that emerged from the water was a creation of the Phoenix's who took Jean's place on the X-Men. In the days following the incident, Jean began calling herself Phoenix and was shown to have greatly increased power. Things then turned tragic in Chris Claremont and John Byrne's "Dark Phoenix Saga." After Mastermind attempted to brainwash her into becoming the Hellfire Club's Black Queen, Jean was corrupted and became the evil Dark Phoenix, who went on a rampage that caused universal chaos, with her even destroying an entire planet. After briefly taking back control of her body, Jean -- the Phoenix-created Jean duplicate, to be precise -- killed herself to save her fellow X-Men and the universe. She would later return to life and eventually regain the power of the Phoenix and although they were able to live with each other better than they had in the past, Jean never achieved complete control over the power.
During the "Planet X" story in Grant Morrison and Phil Jimenez's New X-Men #150, Wolverine stabbed Jean to unleash the Phoenix in the hopes that it would save them from hurtling into the sun. It did so and helped the X-Men defeat Magneto, but not before he hit Jean with an electromagnetic pulse that killed her.
The adult Jean remained dead for a surprisingly long time before returning in Phoenix Resurrection. As the X-Men must deal with the Phoenix Force's return to Earth, Jean is alive but unaware of her history. She lives in a kind of altered reality that presents her with an ideal version of her life, which turns out to be a Phoenix Egg. The team realizes the Phoenix brought Jean back and is trying to mold her into a perfect host before bonding with her once more. After Old Man Logan forces her to realize that her world is an illusion, Jean rejects the Phoenix and convinces it to move on from her.
Although the Phoenix does leave and Jean is currently making an impact on the world all herself as a leading citizen of Krakoa, she has been shown to still have some sort of connection to the entity if only a minor one. When the Phoenix again recently returned to Earth, Jean told Wolverine she felt it, which resulted in him setting off to kill it.
The misconception that Jean and the Phoenix are irrevocably linked is perpetuated by the character's portrayal in other media. In the live-action film X-Men: The Last Stand, Jean, played by Famke Janssen, is brought back to life by the Phoenix, who soon after goes on an increasingly destructive rampage. 2019's Dark Phoenix presented a relatively more comic-accurate version of the story, in which a younger Jean Grey is exposed to the Phoenix Force, which is an independent entity, in space just as in the comics. The alien Vuk also states that only Jean is capable of serving as host to the Phoenix without being destroyed.
Given the Phoenix's long-stated interest in Jean Grey, it's likely only a matter of time before the two cross paths again. But for the moment, Jean is finding herself on Krakoa, and the Phoenix is seeing other people in "Enter the Phoenix."