This is a question that requires quite a complex answer to respond to it. It is important to realise that the relationship between Timothy and Phillip develops during the course of this story. Phillip initially is somebody who, as a product of his white, privileged upbringing, finds it very difficult to be with Timothy, who, as a black man, is considered by Phillip to be beneath him. However, it is key to focus on the way in which Phillip overcomes his prejudice and also the way in which Timothy firmly cares for Phillip and challenges his racial assumptions.
Timothy is always very patient of Phillip. At the beginning of the book, Phillip is immature and basically rude. He treats Timothy with no respect at all, and he behaves very recklessly as well as treating Timothy badly. However, Timothy seems to realize that Phillip is young and will learn, and he remains calm, polite, and patient. He also shows Phillip boundaries, slapping him when he goes too far. It does not take long for Phillip to begin to depend on Timothy, and accept his teachings. By the end of the book they have become friends, until Timothy dies. Even then, he has left a lasting impression and Phillip is forever changed from the experience.
Phillip is a young boy whose American father works for an oil company in the Dutch West Indies. He has been brought up to mistrust black men, primarily because of his Southern-born mother. When he and Timothy, a native St. Thomian sailor, are forced to share a raft together following the sinking of their ship, Phillip learns to depend on Timothy for warmth, food and vision. Timothy serves as a surrogate father for Phillip during their months on the island, and a teacher as well, preparing Phillip for surviving on his own when Timothy is no longer alive to take care of the boy.
Their relationship started when the boat sank and they were onboard together with a cat. Phillip got blinded so Timothy took care of him and protected him until he died.