Ajax是使用客户端脚本与Web服务器交换数据的Web应用开发方法。这样，Web页面不用打断交互流程进行重新加裁，就可以动态地更新。使用Ajax，用户可以创建接近本地桌面应用的直接、高可用、更丰富、更动态的Web用户界面。
3、jQuery
定义：
jQuery是一个快速的，简洁的javaScript库，使用户能更方便地处理HTML documents、events、实现动画效果，并且方便地为网站提供AJAX交互。
特点：
轻量级、链式语法、CSS1-3选择器、跨浏览器、简单、易扩展；
jQuery是一种独立于服务器端代码的框架，独立于ASP.NET或者JAVA；
jQuery是当前很流行的一个JavaScript框架，使用类似于CSS的选择器，可以方便的操作HTML元素，拥有很好的可扩展性，拥有不少插件。
描述：
对于程序员来说，简化javascript和ajax编程，能够使程序员从设计和书写繁杂的JS应用中解脱出来，将关注点转向功能需求而非实现细节上，从而提高项目的开发速度。
对于用户来说，改善了页面的视觉效果，增强了与页面的交互性，体验更绚丽的网页物资。
javaScript框架实际上是一系列工具和函数。
二、三者的关系
下面我用一张导图来阐述这三者的关系：
解释：
javaScript是用于Web客户端开发的脚本语言，Ajax是基于JS语言，主要组合JS、CSS、XML三种技术的新技术，是用于创建交互式网页应用的网页开发技术。jQuery是JS的框架，基于JS语言，集合Ajax技术开发出来的JS库，封装JS和Ajax的功能，提供函数接口，大大简化了Ajax，JS的操作。
简单总结：
1、JS是一门前端语言。
2、Ajax是一门技术，它提供了异步更新的机制，使用客户端与服务器间交换数据而非整个页面文档，实现页面的局部更新。
3、jQuery是一个框架，它对JS进行了封装，使其更方便使用。jQuery使得JS与Ajax的使用更方便
详细情况：
Actually only one of them is a programming language.
Javascript is a programming language which is used mainly in webpages for making websites interactive. When a webpage is parsed by the browser, it creates an in-memory representation of the page. It is a tree structure, which contains all elements on the page. So there is a root element, which contains the head and the body elements, which contain other elements, which contain other elements. So it looks like a tree basically. Now with javascript you can manipulate elements in the page using this tree. You can pick elements by their id (getElementsById), or their tag name (getElementsByTagName), or by simply going through the tree (parentNode, firstChild, lastChild, nextSibling, previousSibling, etc.). Once you have element(s) to work with you can modify them by changing their look, content or position on the page. This interface is also known as the DOM(Document Object Model). So you can do everything with Javascript that another programming language can do, and by using it embedded into wepages you also get an in-memory Object of the current webpage by which you can make changes to the page interactively.
AJAX is a technique of communication between the browser and the server within a page. Chat is a good example. You write a message, send a message and recive other messages without leaving the page. You can manage this interaction with Javascript on the client side, using an XMLHTTP Object provided by the browser.
jQuery is a library which aims to simplify client side web development in general (the two other above). It creates a layer of abstracion so you can reuse common languages like CSS and HTML in Javascript. It also includes functions which can be used to communicate with servers very easily (AJAX). It is written in Javascript, and will not do everything for you, only makes common tasks easier. It also hides some of the misconceptions and bugs of browsers.
To sum up:
Javascript is a programming language (objects, array, numbers, strings, calculations)
AJAX and jQuery uses Javascript
jQuery is for simplifing common tasks with AJAX and page manipulation (style, animation, etc.)
Finally, an example just to see some syntax:
// page manipulation in javascript
var el = document.getElementById("box");
el.style.backgroundColor = "#000";
var new_el = document.createElement("div");
el.innerHTML = "<p>some content</p>";
el.appendChild(new_el);
// and how you would do it in jQuery
$("#box")
.css({ "background-color": "#000" })
.append("<div><p>some content</p></div>");