In most computer languages, every function has a name; the idea of a
function without a name is nonsensical. In Lisp, a function in the
strictest sense has no name. It is simply a list whose first element is
lambda, a byte-code function object, or a primitive subr-object.
However, a symbol can serve as the name of a function. This happens when you put the function in the symbol's function cell (see section Symbol Components). Then the symbol itself becomes a valid, callable function, equivalent to the list or subr-object that its function cell refers to. The contents of the function cell are also called the symbol's function definition. The procedure of using a symbol's function definition in place of the symbol is called symbol function indirection; see section Symbol Function Indirection.
In practice, nearly all functions are given names in this way and
referred to through their names. For example, the symbol car works
as a function and does what it does because the primitive subr-object
#<subr car> is stored in its function cell.
We give functions names because it is convenient to refer to them by
their names in Lisp expressions. For primitive subr-objects such as
#<subr car>, names are the only way you can refer to them: there
is no read syntax for such objects. For functions written in Lisp, the
name is more convenient to use in a call than an explicit lambda
expression. Also, a function with a name can refer to itself--it can
be recursive. Writing the function's name in its own definition is much
more convenient than making the function definition point to itself
(something that is not impossible but that has various disadvantages in
practice).
We often identify functions with the symbols used to name them. For
example, we often speak of "the function car", not
distinguishing between the symbol car and the primitive
subr-object that is its function definition. For most purposes, there
is no need to distinguish.
Even so, keep in mind that a function need not have a unique name. While
a given function object usually appears in the function cell of only
one symbol, this is just a matter of convenience. It is easy to store
it in several symbols using fset; then each of the symbols is
equally well a name for the same function.
A symbol used as a function name may also be used as a variable; these two uses of a symbol are independent and do not conflict. (Some Lisp dialects, such as Scheme, do not distinguish between a symbol's value and its function definition; a symbol's value as a variable is also its function definition.) If you have not given a symbol a function definition, you cannot use it as a function; whether the symbol has a value as a variable makes no difference to this.
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