The game involves players controlling Spider-Man to scale a building using his web shooters in order to defuse bombs planted by the Green Goblin. Spider-Man is only able to use his web lines to move up the building, and if touched by enemies they will cut the line and send him plummeting below, though the player can shoot another web in time to save themselves.
Upon reaching the top of the building, Spidey must then face off against the Goblin, who has planted a "super bomb", whose fuse is activated upon defeating a certain number of enemies and/or defusing a certain number of small bombs. Once the fuse is lit, there is a limited time to reach the super bomb before it detonates, claiming one of Spidey's lives. Upon defeating the Goblin (which involves simply avoiding him) and defusing his super bomb, the game starts over at the next level, which may feature faster-moving and/or more instances of the Green Goblin, taller buildings, taller scaffold sections or a quicker-depleting supply of web fluid. The building is also sometimes a different color.
Source: Wikipedia, "http://en.wikipedia.org/wiki/Spider-Man_(Atari_2600)," available under the CC-BY-SA License.