How to implement AJAX in Share Point 2003 ?

There is no straightaway thing to accomplish this. Although Microsoft has released its AJAX toolkit, it is only compatible with .net framework 2.0 while SP2003 is built on .net framework 1.1. To get around this, we have to go for some javascript libraries.
Anthem.net is the best best to achieve. This is basically an open source with rich control functionality built on the top of microsoft's standard web controls. It supports both versions of .net frameworks as well.
To get the ajax functionality for a set of controls in a web page, add anthem panel and embed all standard web controls within it.

Configuration

To get the functionality for a set of controls in the web page, use Anthem Panel control and embed the controls within it. Or else use individual Anthem Controls.

<anthem:panel id="MenuPanel" runat="server" AddCallBacks="true" AutoUpdateAfterCallBack="true"
TextDuringCallBack="Please wait..." width="100%">

In the above configuration, "Please wait..." would be displayed in the control's Text portion while callback happens (TextDuringCallBack). For example, on button click, button text will change to "Please wait.." and remain till callback finishes. AutoUpdateAfterCallBack ensures that controls are refreshed after callback.
While porting the anthem technology to SP server, there are times when the callback updates doesn't happen properly. In that case, in the button click events (triggering events), add MenuPanel.UpdateAfterCallBack=true;

Now develop the webpart and call the webpages as .ascx.

It works great! Enjoy!