As with any Dickens novel, A Christmas Carol is laden with symbols. The first is in the novel's title; the idea of a carol is, of course, symbolic of Christmas and the good news of hope, joy and redemption. The fact that the novel is divided into five staves rather than chapters reinforces this symbolism. Through this, Dickens draws a parallel between his story and the tradition of Christmas music. The only Christmas Carol that is specifically mentioned in the novel is God Rest Ye Merry Gentlemen, the lyrics of which repeat the phrase Oh tidings of comfort and joy. These words are deeply relevant to the redemptive transformation of Scrooge, and Dickens invites his readers to consider this story as a message of hope and joy and the prospect of a life transformed for the better.
The weather at the story's beginning is also symbolic - especially when contrasted with the weather in the novel's resolution. The dark, creeping, invasive nature of the fog helps to characterise Scrooge himself as we first encounter him isolated in his counting house. The fog symbolizes Scrooge's inability to see clearly the state of his soul. In contrast, the final stave describes the weather as having "No fog, no mist; clear, bright ... golden sunlight." Scrooge at this point in the novel can now see clearly and immediately begins to turn his life around.
Another key symbol in the novel is the chain that Marley wears. We are told that it is made up of "cash-boxes, keys, padlocks, ledgers, deeds, and heavy purses wrought in steel." These items reflect the concerns of Marley's (and Scrooge's) heart and the fact that he wears them as a heavy chain symbolizes the idea that he is now bound by the burdens of his life.
That should get you started on your hunt for symbols in the novel. As you go, look out for things that symbolize the following ideas: redemption and transformation, responsibility (for the poor, for family - consider the Cratchit family), the supernatural (consider especially the symbolism in each ghost's appearance), poverty (both financial and moral), and isolation (look at Scrooge's house).
All the best.