step 1:
get jquery. all you have to do is download the latest file and include it on your page.
step 2:
make 2 files:
somepage.html:
<script type='text/javascript' src='jquery.js'></script>
<script type='text/javascript'>
$.get("someScript.php",
// data to send if you want
{
'someVar' : 'someValue'
},
// receive and do something with response
function(response){
alert(response);
} // function (response)
); // .get()
</script>
someScript.php
<?php
echo $_GET['someVar'] . " response!";
?>
step 3:
upload all your files to your server and go to somepage.html
That's all there is to it. Though, you would generally put that code inside some kind of onclick or whatever, depending on what you want to use ajax for. But the comments in there are pretty self explanatory. jquery is used to make the ajax request, with an example of sending data to the server-side script receiving the request (using GET method). You would do whatever in someScript.php but in this example, it simply echoes back the value you sent. Then jquery takes what someScript.php echoes out and just throws it in a popup.