You should use the SSI-function.
There is several ways but this can solve your problem.
<!--#include virtual="PathToYourFile/YourFile.html" -->
This can be inserted into a <div> for further styling in CSS.
REMEMBER! Due to some limitations in html-doctypes you cannot inlude a .html-file into an .html-file. You have to use another format as .shtml where you can inlude your .html-files. You can include .html into your .shtmlfile. This was also what .shtml was originally created for.
This is because it is part of the XHTML (Dynamic XML HTML)...
To change a file
Your approach on the HTML is correct and also your JS. I include a lot of html-files containing texts there.
My approach is that when a page is loaded some text will be loaded with the <!--#include virtual="" --> inside a <div>. Below JS is used to change the content in the <div>. As Daniel Beck stated below: "...at least in Apache the server needs to be configured to check particular file extensions...".
You configure your file in your .htaccess-file. But ONLY do this if you know what you are doing.
Some (newer?) servers have a default setup of which you don't need to alter the .htaccess-file if you want to be able to include .html-files. At least you are able to include .html-files into .shtml-files.
I have included a Mimetype converter which tells the browser how it should read the file. For txt/html I have told the script that it should use the character encoding ISO-8859-1. Others as UTF-8 could also be used. This depends on your and your receivers native language.
Take into consideration to use the e.preventDefault();. With this i tells the browser NOT to see this as navigation link and will therefore only load the content in the <div>.
$(function() {
  $('#ButtonsID').click(function(e) {
    $('.DivClass').load('PathToFile/File.shtml');
    e.preventDefault();
  });
});
$.ajaxSetup({
  'beforeSend': function(xhr) {
    xhr.overrideMimeType('text/html; charset=ISO-8859-1');
  }
});