The alert box is a system object, and not subject to CSS. To do this style of thing you would need to create an HTML element and mimic the alert() functionality. The jQuery UI Dialogue does a lot of the work for you, working basically as I have described: Link.
<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>jQuery UI Dialog - Default functionality</title>
<link rel="stylesheet" href="//code.jquery.com/ui/1.12.1/themes/base/jquery-ui.css">
<link rel="stylesheet" href="/resources/demos/style.css">
<script src="https://code.jquery.com/jquery-1.12.4.js"></script>
<script src="https://code.jquery.com/ui/1.12.1/jquery-ui.js"></script>
<script>
$( function() {
$( "#dialog" ).dialog();
} );
</script>
</head>
<body>
<div id="dialog" title="Basic dialog">
<p>This is the default dialog which is useful for displaying information. The dialog window can be moved, resized and closed with the 'x' icon.</p>
</div>
</body>
</html>
Answer from PCasagrande on Stack OverflowThe alert box is a system object, and not subject to CSS. To do this style of thing you would need to create an HTML element and mimic the alert() functionality. The jQuery UI Dialogue does a lot of the work for you, working basically as I have described: Link.
<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>jQuery UI Dialog - Default functionality</title>
<link rel="stylesheet" href="//code.jquery.com/ui/1.12.1/themes/base/jquery-ui.css">
<link rel="stylesheet" href="/resources/demos/style.css">
<script src="https://code.jquery.com/jquery-1.12.4.js"></script>
<script src="https://code.jquery.com/ui/1.12.1/jquery-ui.js"></script>
<script>
$( function() {
$( "#dialog" ).dialog();
} );
</script>
</head>
<body>
<div id="dialog" title="Basic dialog">
<p>This is the default dialog which is useful for displaying information. The dialog window can be moved, resized and closed with the 'x' icon.</p>
</div>
</body>
</html>
I use SweetAlert, It's Awesome, You will get lots of customization option as well as all callbacks

swal("Here's a message!", "It's pretty, isn't it?");

Videos
The problem is you're using the alert function:
alert("Please Fill Up Required Field");
You can't style this - it's a browser-generated box. You need to use a modal box, and then you can style it.
In the W3schools example, they are not adding the CSS attributes to the alert element, but to a class on a div:
<div class="alert">
I'd recommend going their route and just making a Div appear to be an alert by changing the CSS.
Try jQuery UI located here: http://jqueryui.com/demos/dialog/
They have a theme roller where you can style the dialog and modal boxes.
-- EDIT --
Answering your second question.
Check out the jsFiddle here: http://jsfiddle.net/8cypx/12/
<link rel="stylesheet" href="http://code.jquery.com/ui/1.8.21/themes/base/jquery-ui.css" type="text/css" media="all" />
<div onclick="check_domain_input()">Click</div>
<div id="dialog" title="Attention!" style="display:none">
Please enter a domain name to search for.
</div>
<script>
function check_domain_input()
{
$( "#dialog" ).dialog(); // Shows the new alert box.
var domain_val = document.getElementsByName('domain');
if (domain_val[0].value.length > 0)
{
return true;
}
$( "#dialog" ).dialog();
return false;
}
</script>
โ
Instead of built-in Javascript dialog box popups - that are ugly and impossible to customize with CSS - I would recommend using jQuery dialog box controls. Those can be styled easily, and jQuery ships with many built-in themes for it, too.
http://docs.jquery.com/UI/Dialog