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Javascript Piece
<script>
jQuery(document).ready(function($) {
    // We'll pass this variable to the PHP function example_ajax_request
    var fruit = 'Banana';
    // This does the ajax request
    $.ajax({
        url: UNIQUE_ID_ajax_object.ajaxurl,
        data: {
            'action':'example_ajax_request',
            'fruit' : fruit
        },
        success:function(data) {
            // This outputs the result of the ajax request
            console.log(data);
        },
        error: function(errorThrown){
            console.log(errorThrown);
        }
    });
});
</script>
PHP Piece
<?php
function example_ajax_request() {
    // The $_REQUEST contains all the data sent via ajax
    if ( isset($_REQUEST) ) {
        $fruit = $_REQUEST['fruit'];
        // Let's take the data that was sent and do something with it
        if ( $fruit == 'Banana' ) {
            $fruit = 'Apple';
        }
        // Now we'll return it to the javascript function
        // Anything outputted will be returned in the response
        echo $fruit;
        // If you're debugging, it might be useful to see what was sent in the $_REQUEST
        // print_r($_REQUEST);
    }
    // Always die in functions echoing ajax content
   die();
}
add_action( 'wp_ajax_example_ajax_request', 'example_ajax_request' );
// If you wanted to also use the function for non-logged in users (in a theme for example)
// add_action( 'wp_ajax_nopriv_example_ajax_request', 'example_ajax_request' );


// Pass ajax url to JS
// bit inside the function where you pass the vars
wp_localize_script( 'NAME_OF_JS', 'UNIQUE_ID_ajax_object',
    array( 'ajax_url' => admin_url( 'admin-ajax.php' ), 'we_value' => 1234 ) );

?>