Active Directory is Microsoft's answer to a database that handles a company's security centrally (although it's normally deployed as a distributed multi-master database and across several sites). It typically knows about a company's user accounts, computer accounts, security groups, and resources that that need to be published (names of servers, printers, other services).
Active Directory authentication would be you supplying a username and password that can be checked against the Active Directory database, so that it can be checked to allow or deny you a resource (web page, printer, network file share, database, etc) that you wish to access.
Integrated Active Directory Authentication is, for example, when you allow your web browser to send that username/password combination on your behalf, automatically. You get access to what you should have (e.g. a pages on company's intranet), without being prompted for username and password every time a security check is required.
I'm sorry it's technical, but that's because it's technical. I've tried to simplify it a little bit for you.
What's AD used for? Well, an AD admin can generally define what users can/can't do, what their computers can/can't do, many other settings and preferences. They can do once, centrally, and the AD does the rest of the hard work (of making sure the right permissions etc are set in the right places). It's all about having one central repository for this kind of info.
AD is is extensible, in that it can be added to. If you install e.g. Microsoft Exchange Server, it adds extra fields to the AD database, (e.g. SMTP email address, home email server, maximum mail size a user may send, that kind of thing) and then again you have one central place to manage your (Exchange this time) information.
AD was introduced with Windows Server 2000, extended and used with Windows Server 2003 and now Windows Server 2008.