What does (interactive) mean in an Emacs Lisp function?

Just to clarify (it is in the quoted docs that Charlie cites) (interactive) is not just for key-bound functions, but for any function. Without (interactive), it can only be called programmatically, not from M-x (or via key-binding).

EDIT: Note that just adding "(interactive)" to a function won't necessarily make it work that way, either -- there could be many reasons functions are not interactive. Scoping, dependencies, parameters, etc.

I means that you're including some code for the things you need to make a function callable when bound to a key -- things like getting the argument from CTRL-u.

Have a look at CTRL-h f interactive for details:

    interactive is a special form in `C source code'.
    (interactive args)

    Specify a way of parsing arguments for interactive use of a function.
    For example, write
      (defun foo (arg) "Doc string" (interactive "p") ...use arg...)
    to make ARG be the prefix argument when `foo' is called as a command.
    The "call" to `interactive' is actually a declaration rather than a function;
     it tells `call-interactively' how to read arguments
     to pass to the function.
    When actually called, `interactive' just returns nil.

    The argument of `interactive' is usually a string containing a code letter
     followed by a prompt.  (Some code letters do not use I/O to get
     the argument and do not need prompts.)  To prompt for multiple arguments,
     give a code letter, its prompt, a newline, and another code letter, etc.
     Prompts are passed to format, and may use % escapes to print the
     arguments that have already been read.

Furthermore it’s worth mentioning that interactive's main purpose is, in an interactive context (e.g. when user calls function with key binding), let user specify function arguments that otherwise could be only given programmatically.

For instance, consider function sum returns sum of two numbers.

(defun sum (a b)
  (+ a b))

You may call it by (sum 1 2) but you can do it only in a Lisp program (or in a REPL). If you use the interactive special form in your function, you can ask the user for the arguments.

(defun sum (a b)
  (interactive
   (list
    (read-number "First num: ")
    (read-number "Second num: ")))
  (+ a b))

Now M-x sum will let you type two numbers in the minibuffer, and you can still do (sum 1 2) as well.

interactive should return a list that would be used as the argument list if function called interactively.

(interactive) is for functions meant to interact with the user, be it through M-x or through keybindings.

M-x describe-function RET interactive RET for detailed info on how to use it, including parameter to catch strings, integers, buffer names, etc.

One of the "gotchas" that this clarifies is that the argument to interactive is actually a kind of mini-formatting language (like for printf) that specifies the following (for the surrounding function's input):

1. schema (number of arguments and their type)
2. source (e.g., marked-region in buffer and/or user input, etc.)

For example,

'r' Point and the mark, as two numeric arguments, smallest first.

means that the interactive-annotated function needs exactly two arguments. e.g. this will work

(defun show-mark (start stop)
  (interactive "r")
  (print start)
  (print stop))

This will break:

(defun show-mark (start)
  (interactive "r")
  (print start))

Wrong number of arguments: ((t) (start) (interactive "r") (print start)), 2