AJAX based submit form and use PHP to get results...

Very easy to use and understand, hope you will find it useful and easily edit it to fit your needs.

libNog is a 9k library to provide somewhat easier browser-independence. It is not yet complete, and will be continuously revised. It grew from this snippet, and is still growing as I find things I use often.
Presently, libNog features:

A way to browser-independently get an XMLHttpRequest object.
A set of simple methods for GET, POST, HEAD, and to include other Scripts.
Simple methods for className control
A completely abstracted form of getElement*
Browser-independent methods for attaching and detaching event handlers
An event-normalization routine
A simple one-command 'Stop the Event' routine
And the big one: A simplified Class object by which classes can be built in JavaScript.

Simple chatting script to be placed on a corner of a website
Also a good example how to use ajax.

This is a brief revision to my previous snippet. It is still what it was: a small, simple system for using XMLHttpRequest without using globals.

I'll put the API spec in the code this time, so as not to clutter the front page further.

Changes:

Changed name to HTTPRequest (far more accurate)
Added HEAD requests
Shortened code by making a GenericRequest function
Used a better method for getting an XMLHttpRequest object (checks multiple MS versions and implements the latest one)

This is a simple script to provide basic cross-platform XML request functionality in Javascript. It's meant to be the core component in any AJAX style framework. It is not an instantiable object, but instead is a namespaced microlibrary.

Calling is easy:
XMLRequest.GET(uri, query, callback, fallback)
uri: The location you're after
query: an associative array of form data to provide via the URL
callback: callback function of the form myCallbackFunction(objXMLHttpRequest), which is called upon successful (response = 200 OK) retrieval of the XML data
fallback: myFallbackFunction(objXMLHttpRequest), which is called upon failed (response != 200 OK) retrieval of the XML data.

XMLRequest.POST(uri, query, form, callback, fallback)
Similar, but does the query using the POST method. 'query' is the URL-appended data, still in associative array form, and 'form' is the same for the POST data.

The basic framework of an Ajax-enabled web page. The following JavaScript shows how to send a request for an XML file and how to receive that request. Of course, in a real life scenario, you'll have to implement better error trapping and actually do something with the XML that gets returned.

I use this in my AJAX as a quick way to get the Value of the selected item in a Select HTML object. If nothing is selected, it returns zero.
(May be basic to y'all, but maybe really usefull to someone else).

A simple AJAX class. The code for the class is provided as well as example HTML and PHP code for usage.

AJAX: In situ edit
A simple edit in place using Prototype, based on the tutorial @ 24ways.org