When we think about characters, we often use E. M. Forster's distinction between "round" and "flat" characters. Flat characters are characters that have one or two characteristics alone and are never fully developed or shown to be more than two dimensional paper dolls. They do not grow, mature or develop during the story. Round characters, on the other hand, are characters that are fully painted - we know their motives, what they think, and in addition, they are normally characters that learn and develop throughout the course of the tale.
In "The Rocking-Horse Winner", then, it is clear that out of the three main characters, Paul, the mother and Uncle Oscar, Paul is clearly a round character. We are given some indication of his psychological complexity in his dealings with his mother. It is clear that he is aware of the subtext of what is happening in his house and embarks on a quest for "luck" to gain his mother's affections:
Absorbed, taking no heed of other people, he went about with a sort of stealth, seeking inwardly for luck. He wanted luck, he wanted it, he wanted it.
The lengths to which he is prepared to go shows that he is a round character.
On the other hand, it appears that both the mother and Uncle Oscar are flat characters, in that they do not change and they have but one or two defining characteristics. The mother seems to be defined by her greed and also her inability to love her children:
She had bonny children, yet she felt they had been thrust upon her, and she could not love them.
This is something that continues throughout the story and could be said to lead to Paul's death as he is driven to ever greater lengths to win her affection. Uncle Oscar likewise appears as an avuncular figure, acting as the go between for Paul and supporting him.
Therefore, Paul is the round character in this story, and the others are flat characters because they do not show any form of development and are not necessarily made into three-dimensional characters.