1. AJAX
Chapters 19, 20

2. What it stands for Asynchronous JavaScript and XML (AJAX, an acronym)
It's still a loose collection of technologies, as DHTML is.
It's not really new--just a new idea
Jesse James Garrett coined the term, so he's considered the "founder" (in this article). Note schema is also in Teague book

3. The core object in the DOM is
As you saw in the Sitepoint video, there are a lot of variations. Often, as in the book something other than XML is used to connect to server-side content, e.g. PHP, ASP, JSP, CFML, SSI
The core object in the DOM is
XMLHttpRequest()
AJAX generally is a good thing, but it limits accessibility and SEO
(There's a section on what Web 2.0 is here, for some reason.)

4. The formula
Action + event + fetch data + server process + filter data (browser processes server response) = reaction (page updated)
Key properties
readyState (status of server request)
responseText (string returned from server)
onreadystatechange (automatic event handler)

5. One basic AJAX example This fetches data from a PHP page
Note inclusion of earlier Microsoft ActiveX object Microsoft.XMLHTTP
Keys again are
XMLHttpRequest()
object.innerHTML
Other examples show POST back to the server, and filtering/error checking
For more, try ajaxpatterns.org