jQuery.post('/someurl.php',
{one:'value1',two:'value2'},
function( response ) { alert(response); } );
On someurl.php, you would get these values with:
$oneValue = $_POST['one'];
$twoValue = $_POST['two'];
echo '{success:1}';
How you save them is then up to the PHP code.
EDIT
JavaScript has specific restrictions from being able to access a local computer. Otherwise any hacker's script would be able to save files to your computer, which we wouldn't want.
You mentioned localhost in your comment above and that sounds like the best approach most likely. Post to the site, get the response, and then post the response to your localhost server where you do have access to the server side code to be able to save it to your computer. Something like:
jQuery.post('/someurl.php', {}, function( data ) {
jQuery.post('http://localhost/save.php', data, function() { alert('saved'); ]);
} );