The radioactive spider is what gave Peter Parker his spider powers after it bit him.
When Peter Parker was in high school he went to a demonstration of the Neogenic Recombinator. The experiment was hosted by Dr. Farley Stillwell who invented the Recombinator. As Stillwell activated the Neogenic Recombinator a small spider got in the path of the Recombinator's radioactive beam which made the spider radioactive. The spider then crawled onto Peter's hand and bit him. The spider's venom was now radioactive so instead of killing Peter the radioactive venom gave Peter spider like powers. The spider also passed on it's knowledge of how to make spider webbing which Peter would use in his Web-Shooters. The spider then died after it bit Peter.
In the comics
In 2001, it was revealed that a mystical being known as the Great Weaver chose the spider to be his avatar. The Great Weaver manipulated the spider into coming into contact with radiation and then had it bite Peter Parker giving him his powers.
Also in 2001, it was revealed that a another student named Carl King witnessed Peter being bitten by the spider. Carl wanted to gain the same powers as Peter so he ate the spider. However, the spider caused Carl's body to break down over a period of time and turned him into a swarm of thousands of spiders sharing a single consciousness.
In 2014, it was revealed that after biting Peter, and before being eaten by Carl King, the spider also bit another student named Cindy Moon. Cindy gained similar powers as Spider-Man and she became a superhero named Silk.
In the Ultimate Comics, OsCorp created a drug called Oz as a way to try and recreate the formula that made Captain America a super soldier. Oz was injected into various animals to test the formula. One of these animals was a spider. This spider got loose and bit Peter Parker on the hand and the effects of the Oz drug mixed with the spider's venom gave Peter spider powers.
In the 2002 Spider-Man movie, the spider that bit Peter was a genetically mutated spider instead of a radioactive spider. This spider was mutated to have the abilities of other spiders.
In The Amazing Spider-Man series, Richard Parker, who was a scientist working for OsCorp, created a series of genetically modified spiders. Richard's research on the spiders was meant to save lives. However, Norman Osborn wanted to use the spiders as biological weapons. When Richard refused he fled the country with his wife, Mary, but Norman had them killed. It was later revealed that Richard had placed his own DNA into the spiders. Because of this when the spider bit Peter Parker his father's DNA mixed with the spider venom gave Peter spider powers.