Peter is Mitch's brother. These two brothers lost connection after Peter moved to another country running away from his family and the disease he has.
Just because of Morrie, Mitch wants to find the brother, to renew their relationships. Peter lives in Spain after having moved to Europe immediately after graduating from high school. If you want to know in details, he is now suffering from pancreatic cancer and flies to various European cities seeking treatment. However, he continually refuses to accept help from his family, namely from Mitch, as he has, for the most part, estranged himself from them after his departure from the United States. He does not want help from Mitch or any other member of his family presumably because he has too much pride to accept it. Growing up, he earned a reputation as the bad family boy, as where Mitch had been the family's clean-cut, straight-A student. Mitch's brother is a man who does not want help from a family he has deserted, and who feels that he must prove himself and his independence to them.
Much like Mitch had during his college years at Brandeis, Peter protects himself with a thick veneer of toughness. He has not asked for help from his family since his high school graduation and has no intention of doing so as an adult. When Mitch contacts him, he is very reluctant to reestablish a relationship with his brother and leaves a curt message that he is doing just fine and does not need anyone else's help. He also reminds Mitch that he does not want to talk about his illness. But as Mitch learns from Morrie, everyone, to some degree, needs other people to survive, thus the quote by Auden which Morrie recites numerous times during his lessons with Mitch, "Love or perish."