I'm really not going to be able to explain it any better than the Wikipedia Article on DOM
But to answer a few of your questions:
Where do we still use it?
Every web browser since the mid-nineties.
Who uses it,
Every web developer since the mid-nineties.
in what technology?
Mostly the web via JavaScript, but pretty much anytime you access XML/HTML programatically you are using some kind of DOM implementation.
How important is it for anyone in .net carrier? [sic]
Extremely, although you probably use it without even knowing it.
Is this just a thing of the past which was heavily used but had problems?
If it is then somebody needs to tell John Resig that he has wasted the past 3 years of his life.
