Do we know what happened to the radioactive spider after it bit Peter Parker?
Did it bite anyone else or did it just live out the rest of it's spider-life in peace?
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In Amazing Fantasy 15, it dies from massive radiation exposure.
But, as the experiment begins, no one notices a tiny spider, descending from the ceiling on an almost invisible strand of web...
A spider whom fate has given a starring, if brief, role to play in the drama we call life!
Accidentally absorbing a fantastic amount of radiation the dying insect, in sudden shock, bites the nearest living thing, at the split second before life ebbs from its radioactive body!
As Jamil points out in another answer here, this was retconned in 2014 to provide the origin story of the newly-introduced hero Silk.
In the movie itself, we see the spider crawl under some equipment in the Oscorp lab immediately after it bites Peter:
Interestingly, this is contradicted by the film's novelization1, where the spider dies immediately:
[Peter] looked down, his eyes narrowing, as he watched what appeared to be a spider flip over onto its back, its legs curling up like something out of a commercial for Raid.
The spider's fate isn't shown on-screen, but it seems most likely that Peter killed it immediately after it bit him:
His hand is right over where the spider bit him and, though we don't actually see it in the movie, at least my reaction to a sharp biting sensation on my skin is to smack the affected area with extreme prejudice.
1 Due thanks to Richard for the quote
After biting Peter Parker it bit Cindy Moon (aka Silk) and then died. Cindy exhibits similar powers to Peter such as wall crawling and spider sense. Interestingly, she can make organic webbing from her finger tips. The two also share a pheromone connection that makes each attracted to the other. Check out Amazing Spider-Man Vol. 3 #4.
The following is an photo taken from this issue. Parker just received secret memories from the Watcher's Eye and can't believe what he's seeing.
Parker: The spider! After it bit me -- it bit someone else?! A girl? I -- never knew! Who is she?
Mrs. Moon: Cindy!
Cindy: Mom! Dad! I can't control it!