From the game box:
He spins a web, swings across the sky and lands with a somersault onto the Daily Bugle building. Who is this masked hero? It's you! Fight, jump and web-sling as Spider-Man as you take on a legion of sinister villains. The city is depending on you. Start swinging...
Basic Plot (adapted from Wikipedia):
Peter Parker witnesses 'Spider-Man' steal an experiment from reformed Doctor Octavius's public demonstration. Eddie Brock (Venom) takes a picture of 'Spider-Man', but 'Spider-Man' shatters his camera. Venom surfaces and swears revenge. The police begin a manhunt for Spider-Man. Two mysterious people unleash a fog upon the streets of New York.
Various other Super-Heroes show up during the game (Black-Cat, Daredevil, Human-Torch, Punisher, Sub-Mariner), but most of them don't actually assist Spider-Man.
Spider-Man foils a bank robbery, beats Scorpion (saving J. Jonah Jameson), outruns the police, catches Venom, defeats Rhino, frees Lizard-Man, rescues his wife (Mary-Jane), defeats an infestation of symbiotes, defeats Mysterio (who was the imposter Spider-Man), defeats more symbiotes, defeats Doc-Ock (he's not really reformed), defeats Carnage, and then outruns a unified 'Monster-Ock'. Spider-Man is rescued from an exploding underwater base by Black-Cat, Captain America, and Venom.
Accomplishing goals (or using cheat codes) unlocks different costumes (Symbiote, Spider-Man Unlimited, street clothes, Quick-Change Spider-Man, Amazing Bagman, Spider-Armor, Scarlet Spider, Ben Reilly, Captain Universe). While some are just character skins, others give new abilities.
What If? mode:
Activated by a cheat code, What If? mode (introduced by the Watcher) changes subtle things in the game, mostly for comic effect. (What If? was a series of comics that looked at alternate takes of Marvel Comics history). The main story plays the same.
Hidden throughout the game are comic-book covers. When collected, the player can read a small blurb about the covers (spanning major events in Spider-Man's career) in a gallery.
* The PlayStation, Dreamcast and Windows versions have pre-rendered cutscenes whereas the Nintendo 64 version shows captioned freeze-frames done comic book style.
Source: Wikipedia, Spider-Man %282000 video game%29 , available under the CC-BY-SA License.
This game pre-dates the Spider-Man Movies, and is NOT the same game as http://videogamegeek.com/videogame/68297/spider-man.