The 'fuzz' refers to the police. It is an example of the youthful slang that colours the story. Other examples of slang used in the book are 'heaters', which means guns, 'tuff', which is an adjective that means 'cool, sharp', 'rumble' which means a gang fight, and of course, the word 'Greaser' which means the gang and (more loosely) the social class that Ponyboy is from. The word Greaser refers to the distinct look of this gang, where the members sport long, greased hair. All these terms are of course reminiscent of the youth culture of the Sixties. It is not surprising that the story should be peppered with this kind of slang as it is told from the perspective of a teenager, a Greaser from the poorer part of town whose gang is in perpetual conflict with the wealthier 'Socs'.
Although the story is obviously rooted in the youth culture of a distinct and distinctive era, the issues that Ponyboy and his friends face are common to teenagers of any time and place: coming to terms with the realities of society, coping with school and planning for the future, figuring out relationships with peers and family and members of the opposite sex. In short, the story is all about young people trying to find their way in the world.
Of course, there is also a lot of added drama in Ponyboy's story of gang rivalries and the constant and very real threat of violence, of the heartache of losing both parents, and various escapades such as running away and hiding out from the police. The book has never ceased to be popular, which emphasizes the fact of its relevance even today, several decades after the 1960s.