The original and potentially still the best, Robert Bruce Banner is the Hulk we all know and love: The man who risked his life to save someone from the Gamma Bomb they'd designed, only to get caught in its blast himself and end up with the most destructive case of Multiple Personality Disorder ever (Because, yes; the Hulk isn't the simple Jekyll and Hyde character he started out to be, but someone who's been diagnosed as having multiple personalities resulting from childhood abuse. Sorry, those who think that kind of thing has no place in superhero comics).
After a particularly destructive rampage, a collection of Marvel's superheroes decided that everyone would be happier if the Hulk was sent to a nice alien planet where he could do whatever he wanted without hurting anyone. Of course, things didn't go plan, and he ended up on a planet filled with enslaved aliens that he helped free, becoming that planet's ruler and, in the process, falling in love with Caiera, who becomes his bride before being killed by the explosion of the spaceship that brought him to the planet. (All of this is from the Planet Hulk storyline, soon to be a DVD animated movie:
In many ways, that trailer hits the highpoints of the story.)
Following the death of his wife, the Hulk got together with a bunch of freed slaves and returned to Earth to punish the heroes who'd sent him to space in the first place, blaming them for the explosion and Caiera's demise. After fighting a lot of superheroes and turning New York's underground into a modern Gladiatorial battle between Iron Man and Mr. Fantastic, it was revealed that it was one of the slaves who'd caused the explosion - He was worried that a happy Hulk was a soft Hulk, and so wanted to make him mad. Never a good move with the Hulk - and the Hulk pretty much let himself get defeated, turning back into Bruce Banner and being contained by the US military (This was World War Hulk).
He was released from captivity to fight the Red Hulk, but ended up having all the gamma radiation in his body drained by his crimson counterpart, meaning that he was stuck as Bruce Banner forever... Or, at least, for a couple of months; while Incredible Hulk #600 made the "Bruce Banner forever" claim in July 2009, it only took until October for Banner to get re-irradiated for future Hulk adventures.
Jennifer Walters became a Hulk when she had an emergency blood transfusion from her cousin Bruce Banner. Unlike her cousin, though, she's had no history of mental illness, and so keeps the same personality when she hulks out, meaning that she's the respectable member of the family. That probably explains why she's been a member of so many superteams, including the Fantastic Four, Avengers and Defenders.
Currently, she's missing, following an explosion. The Red She-Hulk has claimed that she's dead, but, come on. It's comics. That never works.