We’ve been here before with Peter Parker, the bright but socially awkward teenager who gets bitten by a radioactive spider and develops extraordinary powers. The last time was 10 years ago, when Tobey Maguire played Parker for director Sam Raimi. But filmmaking technology has made tremendous leaps over the ensuing decade, while the genre of comic-book superhero movies has grown ridiculously cluttered.

Part of the reason “The Amazing Spider-Man” feels so fresh and invigorating is that its story is so simple — anyone remember exactly what the deal was with Loki and that cube? — and its protagonist so relatable. Instead of a billionaire playboy or a Norse God who can beckon thunder, you get a teenager trying to survive adolescence just like everyone else, except his hormones aren’t entirely human, and his puberty will be more torturous than most.

As Parker, Andrew Garfield (“The Social Network”) looks just young enough to pass for 17, and he sells the illusion with a terrific physical performance: He’s all wild arm movements and impatient leg twitches, constantly thrumming with a pent-up energy that will serve him well in short time.

The screenplay, by James Vanderbilt (“Zodiac”), Alvin Sargent (“Ordinary People”) and Steve Kloves (the “Harry Potter” series), refashions Peter from endearing square to a brainy, self-aware outcast who has never gotten over being abandoned by his parents. He doesn’t let anyone get emotionally close other than the uncle (Martin Sheen) and aunt (Sally Field) who have raised him. So when Peter falls for the beautiful Gwen Stacy (a hugely appealing Emma Stone), and she seems to reciprocate his interest, you share his enthusiasm and excitement, the sugary thrill of that ever-important First Love.

“The Amazing Spider-Man” was directed by Marc Webb, whose previous film “(500) Days of Summer” turned out to be a perfect practice run for the Peter-Gwen romance. The movie involves you with this sweet, likable couple, and some of the film’s best beats belong to them (such as a scene in which Gwen must keep her father, played by Denis Leary, from realizing Peter has snuck into her bedroom). But then there’s the matter of that spider, and a well-intentioned scientist (Rhys Ifans) who is experimenting with reptile DNA in hopes of someday growing his missing arm back.

The movie takes its time gradually hooking you on an emotional level. It draws you close: Then, mayhem.

Webb shot “The Amazing Spider-Man” using 3D cameras, and he has composed his images with the giant IMAX screen in mind. In the film’s big set pieces, he often trains the camera on the spot where the characters will end up instead of chasing after them in a blur. The action is clear and easy to follow but also imaginatively staged, like comic-book panels (the sequence in which Parker discovers his powers while riding a subway car is a marvel of choreography and editing). There are several point-of-view shots of Spider-Man swinging through the canyons of Manhattan where the 3D gives you a giddy, elevating rush, and even though there’s only one villain — a giant lizard-monster — the picture’s climax is infinitely more exciting than watching a bunch of superheroes battling hordes of faceless invaders from outer space.

Despite its enormous size — this is an expensive-looking movie — “The Amazing Spider-Man” always remains intimate in scope, with Peter and Gwen front and center. That’s the same thing Stan Lee and Steve Ditko did when they created the character on the page: They kept the stakes personal, so they actually matter. Yes, you’ve seen this origin story told before, but never like this, and not with Gwen Stacy, either. Comic-book readers who know what the future holds will be particularly appreciative of the way in which Webb sets up what is to come. If you don’t know, that’s even better.

The question that’s been hanging over “The Amazing Spider-Man” since the cameras started rolling was whether it was too soon to reboot the franchise. Did we really need another one? As long as it’s this much fun — yes, yes, we do.