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Well, before this my knowledge of Spider-Man mainly came from a 1960s cartoon that I saw growing up in the 80s (unaware at the time that it was a couple decades old). I'm afraid I never really read Spider-Man comic books, so for the most part I can't say how closely the movie sticks to the story, but it seemed pretty good to me. The only major difference I'm aware of is that once Peter acquired his powers, they included generating his webs, whereas in other incarnations, his web fluid was something he created artificially. And while I don't see any real need to have changed that for the movie, it didn't bother me at all, either. I gotta say, it really is an incredibly cool movie. A great story, well written, with good acting, great special effects and action, plenty of nicely subtle (as well as overt) humor, and great dramatic moments, both large-scale and personal.
It begins, of course, with a high school science nerd named Peter Parker (Tobey Maguire), who is also an aspiring photographer. He gets picked on a lot, and doesn't really seem to have any friends except for Harry Osborn (James Franco), who happens to be the son of a rich industrialist named Norman Osborn (Willem Dafoe), head of Oscorp Industries. Norman is often disappointed by his son, but he likes Peter, so I reckon Harry is a bit jealous about that. Another important person in Peter's life is Mary Jane "MJ" Watson (Kirsten Dunst), who's lived next door to Peter since they were young children. He's always loved her, but, while she seems generally sympathetic toward him, for the most part she barely seems to know he exists. She's currently dating a jerk named Flash Thompson.
One day their class goes on a field trip to a lab where super-spiders have been created. Well, really they're just a new species of spider that was created by combining genetic material from a few different species of spiders, each of which had its own special abilities, so the super-spiders have all those abilities. And Peter ends up getting bitten by one of them. (Actually, I guess this is another departure from the original story, in which it was, I think, just an ordinary spider that happened to have been accidentally exposed to radiation, before it bit Peter. But again, this change isn't really important.) The next day, Peter woke up with a bunch of new powers of his own. The first thing he noticed was that he no longer needed his glasses. He'd soon discover he had greater strength, speed, and reflexes, as well as a "spider sense" that alerts him to imminent danger; plus the web thing, which has various uses, such as swinging around the city like Tarzan in a jungle; plus the ability to make incredible jumps; plus the ability to cling to walls and such.
Meanwhile, Norman Osborn was having some problems. Oscorp had been developing this really cool new glider for the military, as well as a gas or whatever that could enhance a person's strength and whatnot. However, the animal testing had, in one case, resulted in side effects of violence and insanity, so it wasn't ready for human testing, and time was running out. Oscorp was in danger of losing its funding to a rival, Quest Aerospace. So Norman experimented on himself. Which, shockingly, turned out not to have been a great idea....
Peter is in need of some cash, so he decides to use his newfound strength in a shady amateur wrestling competition. But he needed a colorful costume. The one he made doesn't quite equal the coolness of the suit he'll eventually start wearing. Anyway, he calls himself "The Human Spider," but the wrestling announcer calls him "Spider-Man." He defeats the formerly undefeated champ, but since he did it in two minutes instead of just surviving for three minutes, the guy in charge of the matches only gives him a hundred bucks instead of the promised three thousand. Because of this, when the guy immediately afterward gets robbed, Peter stands aside and lets the robber escape.
It's important to note that Peter lives with his Uncle Ben and Aunt May, who have raised him. And just before his wrestling match (which his aunt and uncle didn't know about; he said he was going to the library), Uncle Ben tried to talk to him about how he'd been acting recently, and gave him the famous advice, "with great power comes great responsibility" (not that Ben had any idea of the kind of power Peter actually had, of course). But Peter wasn't in the mood to listen, and responded rather rashly. And after his match, he finds that his uncle has been killed by a carjacker, who Peter pursues. And when he catches him, he finds it's the same man he let escape earlier. So now Peter has terrible guilt over that, as well as for what he had said to his uncle. This is pretty much the traditional backstory for Spider-Man, so after all this set-up, the movie's story really begins. And so, it's really after this point that anything I say will truly be spoilers (though if you're entirely new to the character, I apologize for the sort-of spoilers up to this point). You may want to stop reading after this paragraph, if you haven't seen the movie. In case you stop here, I'll just say that Peter moves to the city and becomes a crimefighter who fights a supervillain called the Green Goblin, and it's all very cool and dramatic. The end.
Well. In case you do want to read on.... Quest Aerospace was conducting a test of their new flight suit or whatever, a demonstration for the military. But they were attacked by some mysterious figure. After that, Oscorp would retake the lead in pursuit of the military contract. Meanwhile, Peter, Harry, and MJ graduate from high school (MJ breaks up with Flash), and they make plans for the future. They all move from the suburbs to New York, where Harry and Peter get an apartment together. Peter was apparently working for someone named Dr. Connors, but he gets fired for always being late. We never actually saw him at work or anything, but it's mentioned, and I thought that was interesting, just hearing the name Connors in passing. I always figured it would set up a future villain. (It didn't in this franchise, though Dr. Connors will be much more important in the eventual reboot, The Amazing Spider-Man.) Meanwhile, MJ was working as a waitress at a diner, while also going on acting auditions, without any success.
Of course, the reason Peter was always late for work was that he had a new mission, not to mention a cool new costume. Taking his uncle's words about great responsibility to heart, he'd become a superhero, fighting small time criminals as much as possible, and becoming a local celebrity in the news. Everyone was fascinated by Spider-Man. Though the editor of the Daily Bugle, J. Jonah Jameson (J.K. Simmons), didn't like anything about Spider-Man, except for the chance to sell newspapers with stories about him. Unfortunately, he mostly painted Spider-Man as a menace, despite the public's love of him. But he wanted pictures of Spider-Man, and Peter was in a unique position to acquire such photos, and made some money selling them to Jameson.
Harry, meanwhile, was going out with MJ, though he didn't tell Peter at first. He felt somewhat guilty about it, since he knew his friend loved her, but... Peter had never made a move, and it didn't look like he was ever going to. And Norman was informed of the shocking news that Oscorp was being bought by Quest, and he himself was forced out. After that, at the Unity Day Festival, a new villain attacked, who Jameson would soon dub "The Green Goblin" in his paper. The Goblin's main target at the festival were the board members of Oscorp, but his attack also imperiled MJ, who was attending with Harry. Spider-Man, of course, shows up to save the crowd, and particularly MJ.
Later, Norman discovers he has a split personality, caused by his reckless self-experimentation. He doesn't seem to remember the couple of times he's made attacks using his new powers (and the glider his company had developed). But it will come as no surprise to anyone watching the movie that he is, in fact, the Green Goblin. And then, the Goblin attacks Jameson at the Bugle, but when Spider-Man shows up, he ends up kidnapping Spidey, instead, and offering him a chance to work together, since they're both special. He leaves him to think about it.
Later, MJ is attacked by some thugs on the street, and Spider-Man once again rescues her. And they have what would become a very famous cinematic kiss. Upside down in the rain. She really starts falling for him at this point, even though she's still with Harry. Later still, Spider-Man is once again confronted by Goblin, and declines his earlier offer, which Gobby doesn't take well. However, later Norman attends Thanksgiving dinner at Harry and Peter's apartment, along with Aunt May and MJ. And comes to the realization that Peter is Spider-Man. Meanwhile, he says some nasty things to Harry about MJ, which gets her mad at Harry. Later, to get to Peter, the Goblin attacks Aunt May, so Peter realizes the Goblin knows his true identity. Meanwhile, Peter and MJ begin to get closer, while Harry begins to get closer to his father. Whom he tells that MJ loves Peter, and Peter has always loved her.
So of course, the Goblin then kidnaps MJ to hurt Spider-Man, leading to a climactic battle. At the end of which, the Goblin ends up revealing his identity to Spider-Man, and as he's dying, he asks Peter not to reveal to Harry about him being the Green Goblin. But when Spidey brings Norman's body back to his mansion, Harry walks in and instantly blames Spider-Man for killing him. Later, at the funeral, Harry will tell Peter that someday, Spider-Man will pay for what he did, but ironically he now thinks of Peter as his only family, and is glad to have him as a friend. MJ, meanwhile, confesses her love to Peter. But, wanting to protect her, he tells her they can just be friends. Ah, the sacrifices a hero must make....
So, that's the recap. I hate giving away so much of the plot, but I don't really know what else to do. It's all sort of essential knowledge going into future installments of the franchise....