-ING form

I was reading some things in English when suddenly I noticed that the author started using some unusual construction, at least for me.

Construction using -ING form with some other words I did not know that they could get together.

For instance:
I appreciate your helping me.
Verb + possessive + - ING:
Fancy your having noticed!
Please excuse his not writing to you.
I can't imagine my mother approving!
It's strange him/his behaving like that.
And so on...

Could you be so kind as to tell us why do all these things happen?