AJAX (Asynchronous JavaScript and XML) is a technique for creating fast and dynamic web pages. It allows web pages to update parts of the page without reloading the entire page. This is done by making a request to the server using JavaScript, and then updating the page with the response from the server.

To use PHP with AJAX, you can create a PHP script that runs on the server and sends a response to the client. The client-side JavaScript code can then use the XMLHttpRequest object to make a request to the PHP script, and process the response to update the page.

Here is an example of a basic AJAX request using the XMLHttpRequest object:

var xhr = new XMLHttpRequest();
xhr.onreadystatechange = function() {
    if (xhr.readyState == 4 && xhr.status == 200) {
        // process the response here
    }
}
xhr.open("GET", "example.php", true);
xhr.send();

On the server side, the PHP script can receive the request and send a response back to the client.

<?php
$response = "Hello, World!";
echo $response;
?>

You can also use the $.ajax() or $.get() and $.post() method of jQuery to make the request. They are more user-friendly and have options such as error handling and success callbacks.

$.ajax({
    url: 'example.php',
    type: 'POST',
    data: {name: 'John', location: 'Boston'},
    success: function(response) {
        // process the response here
    }
});

On the PHP side, you can access the data sent via the $_POST or $_GET superglobal arrays.

<?php
$name = $_POST['name'];
$location = $_POST['location'];
echo "Hello, $name from $location";
?>

It is also worth mentioning that in order to use AJAX you must make sure that you are running your PHP script on a server, otherwise you won’t be able to use AJAX calls.

These are just a few examples of how to use PHP with AJAX, you can use many other methods like JSON, serialize, etc. It’s recommended to consult the official PHP and JavaScript documentation for more information on working with AJAX in PHP.