What is Active Directory (AD) and how it works ?

Hello,

I downloaded Server 2008 R2 trial and want to try something.

I found on YouTube that is first step of configuration is that to configure Active Directory Domain Controller (forests and etc.)

First, I want to know what is AD, AD DC, how it works ?

Do I need real domain to make it or can use something like addc1.local ? Can I use it on other computers ?
If answer is yes on second : can I use it outside network, to share it via ex. Hamachi ? Or if answer is no : Can I use hamachi for it or must have real domain and how ?

Thanks in advance.