A spider got caught in a radiation test, the radiation was slowly killing it. Before it died it bit a student by the name of Peter Parker. The spider's bite caused changes to Peter's DNA that gave him certain powers of a spider such as super strength and agility, the ability to climb on any surface, shoot spider-webs using devices of his own invention called "web-shooters," and react to danger quickly with his "spider-sense", enabling him to combat his enemies.
Also attending the exhibit was a student by the name of Carl King, who witnessed Parker being bitten and connected it with the later emergence of Spider-Man. Seeking to gain the same abilities as Peter, King decided to eat the spider, only for a completely different result to occur. His body began to break down over time, eventually metamorphosing him into a swarm of 1000 spiders sharing a single consciousness.
Somehow the spider had time to bite another person inbetween those events, Silk, who got the same abilities as Peter Parker, only better, for some reason.
In other realities other pepole has gotten bitten by the spider, including Aunt May, Gwen Stacy, Flash Thompson, Betty Brant and Bruce Banner