To add to cidbently's excellent answer, it might be helpful to define allegory as a story that uses symbols to tell simultaneously a more general (symbolic) story. We use allegories and symbols all the time.
Parables and fables are short allegories, for example. Think about the story of the prodigal son from the bible or the story of the fox and the sour grapes from classical collections of fables. When people today say "killed the fatted calf" or "sour grapes," they're thinking of these stories, but they're remembering them not just on the literal level (the level of the actual story) but also on the symbolic level (where "sour grapes," for example, means something like "pretending to yourself and others that you do not want now what you wanted just a short time ago because you couldn't get it").
Not all stories that use symbols are allegories, of course, and allegories are often viewed critically today. They're often seen as simplistic or moralistic by many modern readers, probably because allegories have a long history of being used for didactic purposes (e.g. see Plato's allegory of the cave in addition to the parables and fables referenced above).
Basically, an allegory is a type of story and symbolism is a tool that is used in virtually every form of literature. When an author uses symbolism, she causes an object to represent something else. For example, a dove might represent peace.
In an allegory, the author tells one story on the surface, but that story has a deeper meaning underneath. One of the most famous examples of allegory, and one that most people are familiar with, is the Chronicles of Narnia series written by C.S. Lewis. In the opinions of many people, these stories are a clear allegory of the story of Christ.
The difference between symbolism and allegory is rather thin. On the one hand, an allegory would have to be a symbolic text, on the other, symbolism is a necessary technique of an allegory.
Both symbolism and allegory relate to a process of representation where one thing stands for another like light representing life and darkness death, to use a cliche.
A text like Waste Land by T.S. Eliot is an allegory of the War-ridden Europe though it hardly refers to the scene directly. Symbolism is a basic technique in the text whereby the themes like alienation and absurdity are addressed for example in the London Bridge section in the first part of the poem.
Symbolism is generally the literary technique or trope or device that facilitates the allegorical layer of a text while allegory is rather the generic model or the genre of the text.
If Coleridge's Kubla Khan is an allegory of the poetic process, then symbolism is the device that has helped set up the allegory and sustain its individual symbols that make up the allegory overall.