In a general sense, a function is a rule for carrying on a computation given several values called arguments. The result of the computation is called the value of the function. The computation can also have side effects: lasting changes in the values of variables or the contents of data structures.
Here are important terms for functions in Emacs Lisp and for other function-like objects.
car or append.  These functions are also called
built-in functions or subrs.  (Special forms are also
considered primitives.)
Usually the reason we implement a function as a primitive is either
because it is fundamental, because it provides a low-level interface to
operating system services, or because it needs to run fast.  Primitives
can be modified or added only by changing the C sources and recompiling
the editor.  See section Writing Emacs Primitives.
command-execute can invoke; it
is a possible definition for a key sequence.  Some functions are
commands; a function written in Lisp is a command if it contains an
interactive declaration (see section Defining Commands).  Such a function
can be called from Lisp expressions like other functions; in this case,
the fact that the function is a command makes no difference.
Keyboard macros (strings and vectors) are commands also, even though
they are not functions.  A symbol is a command if its function
definition is a command; such symbols can be invoked with M-x.
The symbol is a function as well if the definition is a function.
See section Command Loop Overview.
t if object is any kind of function,
or a special form or macro.
t if object is a built-in function
(i.e., a Lisp primitive).
(subrp 'message)            ; message is a symbol,
     => nil                 ;   not a subr object.
(subrp (symbol-function 'message))
     => t
t if object is a byte-code
function.  For example:
(byte-code-function-p (symbol-function 'next-line))
     => t
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