Peter Bishop (Joshua Jackson) was originally contacted by the FBI for help in having his father, the brilliant fallen scientist Walter Bishop (John Noble) released from a mental institution. Walter's extensive expertise with "fringe-science" was desperately needed to save an agent's life. After getting his father out of the hospital, Peter continued to help the FBI keep Walter focused, coherent, comfortable, and sane. At first he did not want to be around his father as they had a troubling history together but as the series has progressed, their relationship has grown much closer.
Peter has stepped up many times and contributed his varying skills to aid the Fringe team far beyond his original role. He is officially a "consultant" for Homeland Security, but has found himself in numerous gunfights, fistfights, and dangerous situations, moving his role far beyond that of a consultant or chaperon. He takes part in every Fringe Division investigation, offering scientific knowledge, participating in interrogations, and helping Walter in his laboratory. Peter is often the one to explain the complicated, confusing, and often mind-bending scientific "mumbo-jumbo" that Walter rattles off periodically. He is also a potential love-interest for FBI Agent Olivia Dunham (Anna Torv), as they have grown very close throughout the series.
Peter is literally from an alternate universe. He was kidnapped by Walter as a small child, after Walter's own version of Peter died from an incurable illness. He had the same illness that Walter's original son died from, but Walter was able to develop a cure. He then crossed over to the alternate universe and took Peter in order to save his life. After Peter came here, Walter could not bear to send him back and decided to keep him. Peter was unaware of his origins until late in season 2, when this knowledge caused him to leave the Fringe Division and Walter to think things through.
Being from another universe is perhaps why Peter is so troubled in this one. While extremely smart, he is a college drop-out with some recent shady history outside of the United States. When working with some organizations in Iraq, he was first picked up by the FBI to help them retrieve Walter.
Peter has managed to stay out of trouble since joining the Fringe Division. Late in season 2, after learning he was not from this universe, he traveled to his home universe. He met his real mother and learned more about his real father who had crossed into this universe to retrieve him. In the alternate universe, Walter is the Secretary of Defense of the United States and behaves quite differently than the Walter that Peter knows. White troubled, Peter is a generally good guy who likes knowing and discovering the truth.
"I've seen strange. This is something else."
"How come when nobody knows and it doesn't make sense, they come to us?"
"Frozen heads don't just get up and walk into places."
"You think we could get the answer to that question without me in the room."