AJAX is the new buzz word that an inventor created. It is just Asynchronous Javascript and XML. The creator thought the original word, XML HTTP REQUESTS was hard to tell to its customers so he formed AJAX, I didn't know why he didn't use XHR but he used AJAX. And that is where the Web 2.0 buzz came up.
Basically AJAX uses the Javascript object XHR and you can send requests to any local script without the page being refreshed, javascript will deal with the request, and response. And javascript will properly present the contents to the screen as a push model without the whole page being distrurbed.
Take a look at Yahoo's AJAX tutorial:
http://developer.yahoo.com/javascript/howto-ajax.html
The define AJAX as follows:
"AJAX web applications are all the rage these days, and one important aspect of AJAX (the "A" and, depending who you ask, the "X") is a mechanism of making asynchronous connections from the back end of a web application with an object called XMLHttpRequest (XMLHTTP for Internet Explorer). This request is used to retrieve XML or other data from web services such as those provided by us here at Yahoo!. In this HOWTO you'll learn how to make web service requests to Yahoo! Web Services with JavaScript and the XMLHttpRequest object."
So Visual Studio is an IDE (Integrated Development Environment) a GUI that will assist the programmer code more efficiently, the IDE has a compiler which will compile your code so you could use. AJAX is just a buzz word as I described earlier, it is just Javascript. Where you send a request, and the response would be a textual XML response.
AJAX.NET is a ASP.NET tools, where an ASP.NET developer could develop AJAX webapplications with ease: http://www.asp.net/ajax/. "It is a free framework for quickly creating efficient and interactive Web applications that work across all popular browsers"
.NET is a Micrsoft Framework http://msdn2.microsoft.com/en-us/netframework/default.aspx . C# is part of the .NET Languages, where it is one of the most powerful highlevel languages. Recently Microsoft delivered .NET 3.5 it has powerful UI Enhancement, data modelling, etc...
AJAX should be used with caution, since it is a client->server->client framework, you have to know that it relies on JAVASCRIPT to grab contents. So it isn't search engine friendly. I would use AJAX where I don't care about search results, I would use it for control panel where an authentication is needed.
Source Control is a way you will manage your Source Code. Such as CVS or SVN, you can collaborate with many people on developing your code. There are many features of Source Control, too much to explain here, but basically it will help you code better with many people.
VSTS is just a Visual Studio Team Suit edition, same as I explained earlier regarding Visual Studio, but this version has Source Control and Team Management software embedded within that IDE. So you will have all at one
Good Luck, AJAX is just 1 function:p So no need to get all hyped up on it :p
I usually call it XHR instead :)
So lets sum up your definitions in a sentence which all will be used:
You use Visual Studio IDE Team Suite to create Web Applications using the .NET framework using C# as our primary language where we will be using AJAX.NET controls to spice up the interface and our team will be using the Source Control that is embedded within VSTS to manage our Source Code so we can all work efficiently and collaboratively.