There are many symbols in this novel. The most obvious one is the scarlet letter A that Hester must wear on her dress. The prison door is a symbol, Pearl, Hester's daughter is a symbol, the red mark on Dimmesdale's chest is a symbol, the meteor, the woods, etc. Once you know what the theme of the story is, and you can read about that here on enotes, you can pretty much figure out what the symbols stand for.
I'll get you started. The scarlet letter A on Hester's chest is supposed to brand her as an "Adulteress" for having a child out of wedlock when not married (at least everyone thinks she is not married - and she is NOT married to the father of the child, in any case). So the scarlet letter stands for sin. However, its symbolism changes as the story progresses. When Hester is out of jail, she becomes an important part of the community and has many skills. So some of the townsfolk say she is very "able" and perhaps now the A stands for that.
It can also be said that the A eventually comes to stand for Hester's individualism. She is pretty brave, don't you think, staying in a place where everyone shuns her for her sin? She is set apart from the Puritan society, but Hawthorne is criticizing the Puritan society, not Hester. So, maybe the "A" stands for "Apart" and being apart from hypocrites is always a good thing.
Try to look at the story and see where the symbols I have outlined appear, and you should be able to figure out what they mean. Good luck!