It's just standard ajax request from web browser.

You can use $.post(..) method of jQuery, to send POST request through ajax, and like u wrote you have to listen on "click" event on every checkbox u have on list.

I would go with something like Input boxes:

<input type='checkbox' data-taskid='4543' class='taskchecker'>
<input type='checkbox' data-taskid='1452' class='taskchecker'>
<input type='checkbox' data-taskid='8532' class='taskchecker'>

Then in javascript:

$(".taskchecker").on("change", function() {
   var taskId = $(this).attr("data-taskid"); // gets task ID of clicked checkbox
   var state = $(this).is(':checked'); // gets if checkbox is checked or not
   
  $.post("/changestatustask", {'idTask':taskId, 'checkboxStatus':state}, function() {
    // you can add some code here in response to ajax request
  );
});

This code is pretty simple and sloppy, and prolly have some mistakes, but should illustrate how can you to do it.