You should avoid using the mysql_* functions as they are deprecated and will not be supported any more! Please consider using PDO instead, as it provides a common way to connect to all types of databases. Mysqli_* functions are an option as well, but that will limit you to just using MySQL.
Here is an example of how to use PDO and get your query working using prepared statements:
// connect.php
$db_host = '127.0.0.1';
$db_user = 'user';
$db_pass = 'pass';
$db_name = 'database_name';
$db = new PDO('mysql:host='.$db_host.';dbname='.$db_name, $db_user, $db_pass);
// the-script-youre-posting-to.php
require('connect.php');
$message = $_POST['message'];
$receiver = $_GET["user"];
$sender = $sendernick["nick"];
// Using prepared statements almost eliminates the possibility of SQL Injection.
$stmt = $db->prepare("INSERT INTO msg(message, receiver, sender) VALUES (:message, :receiver, :sender)");
$stmt->bindParam(':message', $message);
$stmt->bindParam(':receiver', $receiver);
$stmt->bindParam(':sender', $sender);
$stmt->execute();
For more information regarding prepared statements, have a look at the PHP manual.