alpha question

How can I make my animations background transparent in node editor?

If there’s no background objects, then it’s transparent by default. If however, there are objects there, and you want them not included in a particular render, you can place them on another layer, and just not include that layer in that particular render-layer. (assuming of course you still want them in some images)

Perhaps this will help:

If not, maybe this Pdf will get you there - look for the part about separating the shadows:

EDIT: meant to say, that I often hear the Premul button mentioned. Perhaps this will do it too.

transparent by default? What do you mean?

I am doing project where I have video clip running in background and I should put my blender model on it. So I should “cut” off the background. How it is possible?

You need to use the blender video sequencer.

Import your footage
Import your Blender scene on top.
As long as your scene does not obscure the background it will have a transparent background.
You could also do it with compositing nodes.

Do a search on either and read up.