Load HTML File Contents to Div [without the use of iframes]

Wow, from all the framework-promotional answers you'd think this was something JavaScript made incredibly difficult. It isn't really.

var xhr= new XMLHttpRequest();
xhr.open('GET', 'x.html', true);
xhr.onreadystatechange= function() {
    if (this.readyState!==4) return;
    if (this.status!==200) return; // or whatever error handling you want
    document.getElementById('y').innerHTML= this.responseText;
};
xhr.send();

If you need IE<8 compatibility, do this first to bring those browsers up to speed:

if (!window.XMLHttpRequest && 'ActiveXObject' in window) {
    window.XMLHttpRequest= function() {
        return new ActiveXObject('MSXML2.XMLHttp');
    };
}

Note that loading content into the page with scripts will make that content invisible to clients without JavaScript available, such as search engines. Use with care, and consider server-side includes if all you want is to put data in a common shared file.

jQuery .load() method:

$("#y").load("x.html");

2019
Using fetch

<script>
fetch('page.html')
  .then(data => data.text())
  .then(html => document.getElementById('elementID').innerHTML = html);
</script>

<div id='elementID'> </div>

fetch needs to receive a http or https link, this means that it won't work locally.

Note: As Altimus Prime said, it is a feature for modern browsers