In Scheme and Racket, a symbol is like an immutable string that happens to be interned so that symbols can be compared with
eq? (fast, essentially pointer comparison). Symbols and strings are separate data types.
One use for symbols is lightweight enumerations. For example, one might say a direction is either
'west. You could of course use strings for the same purpose, but it would be slightly less efficient. Using numbers would be a bad idea; represent information in as obvious and transparent a manner as possible.
For another example, SXML is a representation of XML using lists, symbols, and strings. In particular, strings represent character data and symbols represent element names. Thus the XML
<em>hello world</em> would be represented by the value
(list 'em "hello world"), which can be more compactly written
'(em "hello world").
Another use for symbols is as keys. For example, you could implement a method table as a dictionary mapping symbols to implementation functions. To call a method, you look up the symbol that corresponds to the method name. Lisp/Scheme/Racket makes that really easy, because the language already has a built-in correspondence between identifiers (part of the language’s syntax) and symbols (values in the language). That correspondence makes it easy to support macros, which implement user-defined syntactic extensions to the language. For example, one could implement a class system as a macro library, using the implicit correspondence between “method names” (a syntactic notion defined by the class system) and symbols:
(send obj meth arg1 arg2) => (apply (lookup-method obj 'meth) obj (list arg1 arg2))
(In other Lisps, what I’ve said is mostly truish, but there are additional things to know about, like packages and function vs variable slots, IIRC.)