Jason Rosser is Eve's younger brother, and was a jail resident.
Jason was in human jail for stabbing a human girl. The girl was found wandering the streets of Morganville and refused to testify against him. Eve however had seen Jason and the girl earlier in the day, and testified against him. It was this testimony that sent him to human jail. He was released in Glass Houses. After being released he began stalking Claire, and later stabbed Shane. It is also implied that he murdered several human girls starting just after his release.
He was involved in an attempt on Claire's life, as an accomplice to Dean. It was implied that he was unhappy with kidnapping her and draining her of her blood and blaming it on vampires so he turned himself in. Frank Collins then comes and rescues her, taking the pitcher of Claire's blood with him.
Jason is an easily controlled child who is always breaking the law.
He successfully staked Amelie at the Feast of Fools, yet lived to brag about it later.
He was unaffected by his father's death, though he showed up to the funeral. He broke into the Glass house, and was chased down the street by Michael Glass, whom he shot in the chest numerous times.
He has a grudge against his sister Eve because she was the main reason he was in jail.
In one of her diary entries at the end of the early books Eve implies that he has used heroin, in Feast of Fools Claire states that he looks like he has used it.
In early books he shows a liking for becoming a vampire, he later achieves this.
It is implied later on in the series that Jason may have developed feelings for Claire