Does Ajax always require the use of node.js?

That is a very broad question, so the answer might be broad as well:

The short answer: Ajax does not require jQuery nor Node.js.

In practice, Ajax is a technology for asynchronous operations utilized by Javascript send data to and retrieve from a server asynchronously(1). Ajax is fully available in plain, vanilla Javascript, and it works as follows (example taken from Wikipedia, see sources):

// This is the client-side script.
// Initialize the Http request.
var xhr = new XMLHttpRequest();
xhr.open('get', 'send-ajax-data.php');
// Track the state changes of the request.
xhr.onreadystatechange = function() {
    var DONE = 4; // readyState 4 means the request is done.
    var OK = 200; // status 200 is a successful return.
    if (xhr.readyState === DONE) {
        if (xhr.status === OK) {
            alert(xhr.responseText); // 'This is the returned text.'
        } else {
            alert('Error: ' + xhr.status); // An error occurred during the request.
        }
    }
};
// Send the request to send-ajax-data.php
xhr.send(null);

This is a classic example, showing both how to use Ajax with vanilla Javascript, and also why it's much easier with other means such as jQuery, shortening the same snippet to just:

$.ajax({
    url: "http://fiddle.jshell.net/favicon.png",
}).done(function(data) {
    // Do something with data.
});

Sources (including vanilla Ajax examples):

• 1. Wikipedia: Ajax
• 2. A Guide to Vanilla Ajax Without jQuery
• 3. jQuery: ajax()