Danny Wilson is a younger brother of James Wilson. Danny started suffering from schizophrenia in his adolescent years. He later attended Princeton University and, even though he was on medication at the time, he still suffered from mild paranoia. As a result, he frequently spent hours on the phone complaining about his "mistreatment" from professors and his roommates, with James, who was in medical school at the time.
James tried to be supportive, but on one occasion he was trying to study for an exam when Danny called. Not having the time, James lost patience and hung up on his brother. The next day, Wilson's mother called him to tell him his brother had run away and left the meds behind. James blamed himself for Danny's disappearance.
When House started working at Princeton-Plainsboro, he found out they needed an oncologist and called Wilson. Wilson jumped at the chance. At first, he spent much of his free time visiting homeless shelters looking for Danny. However, apart from one sighting of Danny while James was sitting in a diner (when James couldn't leave fast enough to catch up to Danny), he has not seen his brother since he disappeared. He eventually gave up looking for him.
Wilson has generally been open to House about his relationships, but House had no idea about Danny until the episode Histories, despite the fact that House knew Wilson's other brother and his parents. House was wondering why Wilson was so concerned about a homeless patient and Wilson finally told him. Wilson, sitting on a step in a rough area of Princeton, tells House this is the last place he'd seen his brother (he doesn't name Danny at this time), nine years ago, putting the date as 1996 or 1995. Later (sometimes contradictory) revelations about Wilson's life, like the timing of his three marriages, make it likely the last time James saw Danny was the "nine years ago", when Wilson would've been in residency. Therefore the day Danny ran away was years earlier, when Wilson was in medical school and before Wilson met House, was circa 1990-1991, during Wilson's first marriage, to Sam Carr.
Prior to the episode The Social Contract, Danny got into an altercation in New York and was taken to the psych ward of New York Mercy Hospital. At that point, hospital staff made the connection between Danny and James and let James know where Danny was. James made arrangements to see him and deliberately lied to and misled House to keep him from knowing why he was willing to miss monster trucks. Eventually, House realized something was up and pressed Wilson on the issue. House knew Danny's name (this is the first time the audience learns it) and used what information he had to investigate someone Wilson had been in contact with, a psychiatric nurse. To Wilson's surprise, House agreed to accompany Wilson to the hospital. Wilson finally confesses the story of Danny running away, why he feels guilty about it, like he can never be "selfish" and turn down someone asking for him, even for one night, despite knowing how irrational this is.
At the critical moment, Wilson went in to see his brother while House once again was on the phone to his team with his latest revelation. The next day Wilson admits - what with Danny adjusting to anti-psychotics again and the many years since they'd seen each other - he and Danny were now strangers starting over again. While Wilson offered to let House meet Danny, it never ended up taking place in the show.