A symbol is a representation of an idea, thing, concept, or quality.
A word, for example, is a symbol of language which references its own definition. That is, it "stands for" some idea, thing, concept, or quality, as defined by its given language. Mathematical symbols likewise represent quantities, operations, et cetera.
It is human beings' ability to manipulate symbols that allows them to explore the relationships between ideas, things, concepts, and qualities -- far beyond the explorations of which any other species on earth is capable.