Jean Grey is one of the original members of the X-Men. Jean is a mutant with telepathic and telekinetic powers. She is one of the most powerful mutants in the Marvel Universe. In the beginning, Jean didn’t have very much power. She was only a telekinetic, and kind of played the damsel in distress a lot for the other members of the X-Men. This greatly changed over time, as she developed her telepathic powers, and reached higher levels of control with her telekinetic powers. On one mission, to save the X-Men, she finally reacher her full potential, and became the Phoenix. What had really happened was Jean became “possessed” by the cosmic power known as the Phoenix Force. Jean eventually became the Dark Phoenix, and tried to destroy the world. She was stopped by her fellow X-Men, and was thought to have died.
Of course, no one stays dead in comics, and it was reconned that it wasn’t Jean Grey at all as the Phoenix/Dark Phoenix, but the Phoenix force itself that was posing as Jean. It was crazy, but that’s what happens in comics when you have a popular character and they die. Got to find some way to bring them back. Jean has been a member of the various X-Men teams, proving herself time and again as being one of the strongest mutants on the planet. She was killed again in a battle with another mutant, Xorn. A younger version of herself has reappeared in the Marvel universe, along with younger versions of the other original X-Men. The have tried to send them back to their own time, with no luck.
Jean has been linked romantically with Scott Summers(Cyclops), who she eventually marries. She also has a mutual attraction towards Wolverine, but they decide not to act on their shared feelings. She has four genetic children, Rachel Summers, Cable, X-Man, and Stryfe. There is a lot of time traveling and alternate time lines in comics, so those four characters are Jean’s kids, but not by birth. The lengths comics go thru to add new characters without aging their established characters amazes me sometimes.
Jean Grey is probably one of the most recognizable X-Men, even thought she has died twice in the comics. She has appeared in other media, though, so she has stayed in the public consciousness. You can read more about her various exploits here.