In the original book, it's 1971 TV special and The Cat in the Hat Comes Back, he is portrayed as a regular and well behaved kid. In the live action film, he is a destructive and misbehaved twelve-year-old. As a running gag, the Cat constantly calls him names other than Conrad.
Conrad had a tight, slightly wrinkled face, slick black hair. He usually wore a hat and a variety of dark coats, always buttoned up. They ranged from dark brown, to a dark green color. He also usually wore black gloves on his hand. He also usually wore black shoes, black shorts or pants and a blue, red or white shirt.
His name wasn't revealed in the book because he's the narrator or in the cartoon because the credits don't list the characters' names. But in the film version, he was given the name Conrad.
In The Cat in the Hat's Learning Library Books, he was given the name "Dick" before the name was later changed to Conrad.