The DOM is the application programming interface for well-defined HTML and XML structures (per W3C's document). It is used in any place where you interact with the elements of a web page (any element - style, text, attributes, etc). You will hear a lot about the DOM with JavaScript and/or JavaScript libraries, such as jQuery (which, of course, is JavaScript). It is also referenced with Java, ECMAScript, JScript, and VBScript.
If you are programming .NET it is important if you are doing web-based work. If you are doing application programming, it's not as important. The DOM is definitely not a thing of the past - it is used and worked within every day by many developers. With that said, there has been work towards standardization of the DOM across web browsers. (Again, libraries can help hide these differences. This is one reason jQuery is so popular. You don't have to worry about the browser specifics - you just do what you need to do.)
The document I linked to above does a great job of answering all your questions and more. I would highly recommend reading it. If you have more questions, you can also check out the links below:
- What is the Document Object Model? (W3C)
- Document Object Model (Wikipedia)