Reload just one part of a webside

I would use AJAX, and specifically I would use jQuery to make AJAX simpler. You don't have to use jQuery though. The strategy is:

1. Give the <td> you want to update a unique ID.
2. Respond to a click event on a button labeled "Update". Respond to this event with jQuery.
3. Use jQuery to call the php script on the remote server.
4. Using jQuery, update the HTML of the <td> using the ID you created in step 1.

Using load() in the first example in this tutorial is probably a good place to start:

5 ways to make AJAX calls with jQuery