javascript variable to wordpress php variable

First you need to to send function name as additional data as

  $.ajax({
        type: "POST",
        url: compareids_ajax.ajax_url,
        data: {                             // Data object
              compareIDs : compareIDs,
              action: 'your_ajax_function'  // This is required to let WordPress know which function to invoke
        },
        contentType: "application/json; charset=utf-8",
        dataType: "json",
        success: function (msg) {
              console.log( msg );
        },
        error: function (errormessage) {
              console.log( errormessage );
        }
  });

In your php file

  add_action('wp_ajax_nopriv_your_ajax_function','your_ajax_function'); // Ajax Function should be hooked like this for unauthenticated users
  add_action('wp_ajax_your_ajax_function','your_ajax_function'); // 

  function your_ajax_function(){

        $comparedIds = $_POST['compareIDs']; // Your data is now available in $_POST 
       // Do your stuff here

    die(); // At the end 

  }

See wp_ajax_(action)