The evolution of Ajax is quite interesting story by itself. The technology has been around for like 10 years but up until Google decided to use it in "Google Suggests" no one had given it a thought.
First a bit of background on what is Ajax - Asynchronous JavaScript and XML. In fact behind this complicated-sounding name stands a very simple concept. Without Ajax the work flow goes like this: user clicks on a link, page loads, end of story. With Ajax however we have a different work flow: once page is loaded in the browser it can execute JavaScript and request additional information from the web server without refreshing the entire page. And that is what makes the technology so cool for websites development - it allows to refresh parts of the page without refreshing the entire browser.
For example: you register on a website and enter your username. In the background (while you enter another data) the page sends that username to the server to check if it is available and notifies you. What you get is that after you entered your username almost instantly you get a warning that it is taken, before you even submitted the entire page. How cool is that?
Based on open standards - Ajax is based on JavaScript, XML (though web developers tend to prefer using JSON instead) and is usually integrated in standard XHTML / CSS pages, so there is nothing fancy. No new standards invented. In fact Ajax powered websites only differ in architecture of the code.
Cross browser - We mentioned before that Ajax has been around for decade and is integrated with absolutely every browser, currently in use. It is however important to know that Ajax cannot be used on machines with JavaScript disabled, but the percentage of those us quite low. Also it is worth pointing out there are quite some differences between the browsers which slow down websites development a lot, but that's not to be afraid of.
Decreases pages loading time - as pointed out Ajax allows fetching information without refreshing. That is important because entire pages often load slowly. Imagine you try to register to popular website. Without the cool username check we used as example you would have to reload the page several times until you hit a free username. And it that page loads like 10 seconds it would be seriously disturbing and would make users give up registering in the first place.
Decreases server / browser load - Ajax allows decreasing both the server and the browser resource usage. For instance let's have a set of drop down lists - first for selecting country, next region and third - city. Without using Ajax you would have to populate all possible values for these lists with the loading of the page. That would be quite a lot of information and required resources (and loading times) will go trough the roof. But if we use Ajax we can load the regions after the country is selected, and will load only those that are in this country. After region is selected we will load via Ajax the cities in this region. True - this sounds like slowing down the user, because he has to wait after each selection, but in fact this will take far less time than waiting for all the information to load at the beginning.
Improves user experience - That is the most important reason. Because you website visitor is the King. And improving the experience is of most importance. By decreasing loading times, providing run-time forms validation (without page refresh), slick gadgets and etc an Ajax-powered website keeps the visitor's interest for much longer.