Since you are new to Javascript development, I'll try with relatable examples.
You can vote questions up or down on StackOverflow. Your vote action is sent to the server, and it gets recorded there. Had it not been for AJAX (and some other techniques), the entire page would need to be refreshed for that one action. AJAX solves the problem of asynchronously communicating with a server without requiring full page reloads.
jQuery is a library that provides convenient access to common Javascript tasks such as DOM manipulation, AJAX handling, etc. jQuery also hides away browser differences and provides a consistent interface for the end user. To illustrate these two points, see these examples:
finding all div elements on the page
// Javascript
var divs = document.getElementsByTagName("div")
// jQuery
$("div")
adding a click event handler to a button (illustrates browser differences)
With pure Javascript, it's best to create a cross-browser method to add events, as you surely wouldn't want to write this code every single time. Source - http://www.scottandrew.com/weblog/articles/cbs-events
function addEvent(obj, evType, fn, useCapture){
if (obj.addEventListener) { // standards-based browsers
obj.addEventListener(evType, fn, useCapture);
return true;
} else if (obj.attachEvent) { // IE
var r = obj.attachEvent("on"+evType, fn);
return r;
} else { // some unknown browser
alert("Handler could not be attached");
}
}
Once this is setup (one-time only), you can add events to any elements using this function.
// Javascript
var button = document.getElementById("buttonID");
addEvent(button, "click", function() { alert("clicked"); }, false);
// jQuery (contains code similar to above function to handle browser differences)
$("#buttonID").click(function() { alert("clicked"); });
AJAX is part of Javascript and not a separate technology in itself. You would use AJAX to avoid doing full page refreshes when you need to send/receive data from the server.
jQuery, MooTools, Dojo, Ext.JS, Prototype.JS, and many other libraries provide a wrapper around Javascript to abstract away browser differences, and provide an easier interface to work with. The question is would you want to do all of this re-work yourselves. If you're not exactly sure what re-work you may need to do, researching pure Javascript examples of common tasks such as AJAX calls, DOM manipulation, event handling, along with abstracting away browser quirks and comparing those to examples to equivalents in libraries such as jQuery might be a good start.