Based on this synopsis and everything we've seen from the film, which admittedly is not a whole lot, one of the biggest aspects of that "untold" quality are the parents of Peter Parker, who is played this time by Andrew Garfield.  In Raimi's films, Peter's parents simply weren't a concern and were only mentioned passingly, but The Amazing Spider-Man seems built around the central mystery of just what happened Richard and Mary Parker, played by Campbell Scott and Embeth Davidtz.

Check the synopsis:

One of the world's most popular characters is back on the big screen as a new chapter in the Spider-Man legacy is revealed in "The Amazing Spider-Man." Focusing on an untold story that tells a different side of the Peter Parker story, the new film stars Andrew Garfield, Emma Stone, Rhys Ifans, Denis Leary, Campbell Scott, Irrfan Khan, with Martin Sheen and Sally Field. The film is directed by Marc Webb from a screenplay written by James Vanderbilt, based on the Marvel Comic Book by Stan Lee and Steve Ditko. Laura Ziskin, Avi Arad, and Matt Tolmach are producing the film in association with Marvel Entertainment for Columbia Pictures, which will open in theaters everywhere in 3D on July 3, 2012.

"The Amazing Spider-Man" is the story of Peter Parker (Garfield), an outcast high schooler who was abandoned by his parents as a boy, leaving him to be raised by his Uncle Ben (Sheen) and Aunt May (Field). Like most teenagers, Peter is trying to figure out who he is and how he got to be the person he is today. Peter is also finding his way with his first high school crush, Gwen Stacy (Stone), and together, they struggle with love, commitment, and secrets. As Peter discovers a mysterious briefcase that belonged to his father, he begins a quest to understand his parents' disappearance – leading him directly to Oscorp and the lab of Dr. Curt Connors (Ifans), his father's former partner. As Spider-Man is set on a collision course with Connors' alter-ego, The Lizard, Peter will make life-altering choices to use his powers and shape his destiny to become a hero.

So Richard Parker and Curt Connors were partners, no doubt on research that will lead directly to Parker acquiring arachnid-like abilities and Connors mutating into a giant lizard-man.  Months ago, unofficial photos from the set showed that Oscorp would appear in the film, and the company's inclusion here lightly suggests that this is probably an element of the story that will be explored in sequels.  Presuming this films performs well enough to actually get sequelized, it's likely we'll see a new iteration of Norman Osborn, previously played by Willem Dafoe.

The Amazing Spider-Man arrives in theaters on July 3rd, and it was actually filmed in 3D, so you might want to grab some Rogue 3D Eyewear before then.