您需要做3件事.

首先，将您的jQuery代码包装在 $(document).ready() 函数中，

First, wrap your jQuery codes inside $(document).ready() function,

<script type="text/javascript">
    $(document).ready(function()
    {
        $('#form_signup').submit(function(event) {
            event.preventDefault();
            $.ajax({
                type: 'POST',
                url: 'signup.php',
                data: $(this).serialize(),
                dataType: 'json',
                success: function (data) {
                    console.log(data);
                    $('#form_signup_text').html(data.msg);
                },
                error: function (data) {
                    console.log(data);
                    $('#form_signup_text').html(data.msg);
                }
            });
        });
    });
</script>

第二，在表单中添加一个提交按钮.另外，您缺少电子邮件输入字段的 name 属性.这会导致php文件中的错误.

Second, Add a submit button to your form. Also you are missing the name attribute for the email input field. That causes the error in the php file.

<form id="form_signup" name="form_signup" method="POST">
    <div>
        <input type="email" id="inputEmail1" name="inputEmail1" placeholder="your@email.com">
    </div>
    <div>
        <input type="submit" name="signup" value="Sign Up!"/>
    </div>
    <div id="form_signup_text">
        <!-- A fantastic notice will be placed here =D -->
    </div>
</form>

第三， echo 个结果，因为您正在使用AJAX提交表单. return 不会有任何影响.

Third, echo the results since you are using AJAX to submit the form. return will not have any effects.

<?php

$our_mail =    "our@email.com";
$subject  =    "Wohoo! A new signup!";
$email    =    $_POST['inputEmail1'];

$return = array();
$return['msg'] = 'Thank you!';
$return['error'] = false;

if(preg_match("/([\w\-]+\@[\w\-]+\.[\w\-]+)/", $email)){

    $message = "Yesss!! We receive a new signup!
    E-mail: $email
    ";
    mail($our_mail, $subject, $message);
}
else {
    $return['error'] = true;
    $return['msg'] .= 'Something is wrong... snifff...';
}

echo json_encode($return);exit;

我检查了，一切正常.

希望这会有所帮助:)