How does Ajax work with PHP?

Question

I'm having troubles using ajax and php. What I'm trying to do is call an ajax function that grabs a value from an form's input, and checks if that email exists in a database. Here is my current javascript:

//Checks for Existing Email
function checkExisting_email() {
    $.ajax({
        type: 'POST',
        url: 'checkExist.php',
        data: input
    });

emailExists = checkExisting_email();

//If it exists
if (emailExists) {
    alert("This email already exists!");
}

Unfortunately, I can't get my alert to go off. In my PHP function, it checks whether the input is a username or an email (just for my purposes, and so you know), and then it looks for it in either column. If it finds it, it returns true, and if not, it returns false:

include ('func_lib.php');
connect();
check($_POST['input']);

function check($args)
{
    $checkemail = "/^[a-z0-9]+([_\\.-][a-z0-9]+)*@([a-z0-9]+([\.-][a-z0-9]+)*)+\\.[a-z]{2,}$/i";
    if (!preg_match($checkemail, $args)) {
        //logic for username argument
        $sql = "SELECT * FROM `users` WHERE `username`='" . $args . "'";
        $res = mysql_query($sql) or die(mysql_error());

        if (mysql_num_rows($res) > 0) {
            return true;
        } else {
            return false;
        }
    } else {
        //logic for email argument
        $sql = "SELECT * FROM `users` WHERE `email`='" . $args . "'";
        $res = mysql_query($sql) or die(mysql_error());

        if (mysql_num_rows($res) > 0) {
            return true;
        } else {
            return false;
        }
    }

}

SO my issue is, how does ajax respond to these returns, and how do I make ajax function accordingly? Mainly, why doesn't this work?

Any help is very much appreciated. Thank you!