Try thinking of a symbol as a word, object, event, or character which represents (symbolizes) something else.
Here are some examples from Fahrenheit 451.
The Mechanical Hound symbolizes the destructive application of technology. The Hound also represents an instrument of war, a tool used by the State to suppress citizens. In other words, the Hound represents the evil application of technology.
Clarisse represents innocence, curiosity, and imagination.
The illegal books (those which the State tries to burn) represent knowledge, truth, and more directly, they symbolize the means (access) by which citizens can obtain truth.
The shows on the parlour walls are also tools of the State. They act like pacifiers. For example, as long as Millie lives vicariously through the lives of the characters, she will not consider the significance of her own life/society. In this way, these shows are like a drug or a sedative in that they pacify a society into becoming thoughtless drones. The shows are indoctrinating propaganda.
The pills Millie takes can also represent a pacifying tool, a means of suppressing thought. The term "opiate" is often used to describe any object or mechanism which an authority uses to keep its citizenry in passive roles.
Near the end of the book, Granger speaks of the Phoenix. In mythology, the Phoenix would live for 500-1000 years, ignite and be reborn or remade from its ashes. The Phoenix symbolizes Montag's rebirth and in general, the potential for a person or society to be reborn from the ashes.
Fire is a complex symbol in the book. Fire is thought to be one of the first technological discoveries. It allowed humans to keep warm in winter, cook food, and eventually it was used for power. In the novel, however, fire is (like the Hound) used as a destructive means of technology. A tool that was once used for progress is here used for destruction of books and free thinkers. But, as the city burns at the end of the novel, fire can also symbolize the chance for rebirth (especially in the section where Granger talks about the Phoenix).
Professor Faber symbolizes a guide or maybe even a father figure for Montag. He introduces some literature to Montag and even speaks to him through a mechanical device in his ear. Here is an example where technology is being used to fuel the fire of Montag's awakening and new thirst for knowledge.
Granger and the book people represent hope for a better, more free future.
The river and the railroad symbolize paths to other ways of life. The river and railroad are literal/physical and figurative paths, geographically leading them to a new place and leading them to a new way of life.
Grr! I'm so tired of people misusing the word "symbol"! While I'm not necessarily disputing that the Mechan. Hound is a symbol, it's not a symbol of the evil application of technology. It *is* an evil application of technology, so it can't be a symbol for it. Faber can't symbolize a guide because he is a guide.