When this unnamed criminal broke into the house of May and Ben Parker, he was frightened by Ben and shot him. The bullet killed Ben, whiched caused his nephew Peter to capture the burglar as Spider-Man, starting his superhero career.
He later returned in an alliance with Mysterio, which included faking the death of Aunt May. The burglar believed an old-time mobster's haul was hidden inside the Parker home, and wanted to force this information out of May and Peter. When Spider-Man appeared to thwart them, Mysterio bailed on the burglar. When an enraged Spider-Man unmasked, the burglar blindly panicked, believing Parker would never let him live. As Spider-Man (who had no intention of crossing that line) chased him down, the burglar died of a heart attack.
Later, in Marvel Comics Presents # 50, Peter, in the company of Wolverine, encountered a bizarre group of criminals, all of whom seemed somehow connected to his past. Peter caught one of them, a doctor of his acquaintance who claimed that Spider-Man had killed his brother, and who was dressed just like the burglar.
During the final days of Ben Reilly's tenure as Spider-Man (when this clone believed that he was the real Peter Parker), Ben met and fell in love with a girl named Jessica Carradine (despite the later movies, this has never been confirmed as his surname) who turned out to be the burglar's daughter. When she learned who her boyfriend was, she threatened to expose him, but came to realize it was her estranged father who had been the villain, and wished him well as she left.
Various retcons and reboots within Marvel's mainstream books have shifted around fine details, but in the "616" universe (The Main Universe), his story remains pretty much the same, even to his final death. Like the radioactive spider itself, the Burglar is a vital element of Spider-Man's origin.
The burglar was seen behind Peter as he was buying a bottle of chocolate milk in a grocery store. He made a mess, causing the clerk to clean up leaving the cash register unguarded. He steals the money from the register and just like the first movie, Peter let him go after he was short-changed on buying chocolate milk and the clerk wouldn't let him take two pennies to pay for it. The burglar ran and bumped into Uncle Ben, who was looking for Peter. Ben saw that he dropped a gun and tried to fight it away from him, but during the struggle the burglar shot him in the chest, killing him almost instantly. Police then gave Aunt May and Peter a sketch drawing of the burglar in case they see him. He has long hair, a goatee, and a yellow star tattoo on his left wrist. It was Peter's hatred of the burglar that caused him to hunt down anyone who looked like him. As of now, he is still at large.
This version of the burglar, like many elements from this film, seems to be based more on the Ultimate Marvel Comics version of the character. This version also has no speaking lines in the film.
In The Amazing Spider-Man 2 video game adaption (which follows an alternative continuity not canon to the films) he is identified as Dennis Carradine (taken from the character's identity in the Sam Raimi films) and is killed by Cletus Kasady.
Being so pivotal a piece of Spider-Man's origin, his character in alternate universes also often changes, both in stories titled 'What If?', and in others that simply show a variant history.
One early What If? story had the newly-arrogant Peter Parker deciding to stop the burglar for the publicity, leading to Spider-Man staying in show business and becoming ever more egotistical.
A four-story anthology focusing on the character of Nova had an embittered Peter Parker, not Richard Rider, receiving the powers and costume of Nova. Racing home to a widowed Uncle Ben as the burglar entered, Peter deflected his gunshot back at him, killing him. Disheartened that even this good fortune had resulted in death, Parker never pursued a superhero career.
Two different What If's had the burglar encountering and killing Aunt May instead, leading to a rocky but ultimately more positive world where Ben was able to guide his nephew through the tough times.
In one story, an angry Peter kills the burglar upon capturing him, carrying the guilt until he turns himself in, serving his time and being released early for not having fully been in his right mind at the time.
One issue of nothing but humorous What If's featured Galactus as Uncle Ben's nephew. Enraged by Ben's death, Galactus restores Ben to life and reduced the burglar to a puddle of protoplasmic goo.