I have heard this sentence in American colloquial language
Ask that assistant of yours.
or
I hate that friend of yours.
Now is the following sentence correct?
I will stay at the place of my friend's.
Does anybody know what type of grammatical structure is this?
This sentence isn't correct. If you want to say this, you'll use:
I will stay at my friend's place.
A popular way to say it is also to remove 'place', leaving:
I will stay at my friend's.
Nonetheless, you could say:
I hate that place of yours.
To distant yourself from it and to show disdain towards it.
The first two sentences you provide are possessives:
Ask the assistant of yours.
I hate that friend of yours.
These are also possessives, and they mean the same thing:
Ask your assistant.
I hate your friend.
But I think what you're asking about are double possessives or double genitives.
A double possessive uses both forms of possessive at the same time. For example:
That book of David's is heavy
At first, this seems like a strange thing to do. Why would you use both possessive forms? The answer is that book of David is considered awkward, but that book of David's is not. This is an old construct dating back to Middle English, and that's why it might seem odd.
But why would you use it instead of using an apostrophe, which is the more common and accepted way? It just gives you a different option for communicating possessives; in some cases it might sound better and make the sentence flow more smoothly. It also draws more attention to the phrase, and it implies a familiarity with the object:
I love that sweet man of mine.
She lives in a world of her own.
There is a rule that states you should not use a double possessive if the object of the prepositional phrase is an inanimate object. So it'd be ungrammatical to say:
The laws of the church's (bad)
Instead, you can say:
The laws of the church
The church's laws
You might want to avoid double possessives, because a lot of them do sound awkward. There's no one rule that determines whether a double possessive sounds good or not. But nobody would say these:
I will stay at the place of my friend's. (bad)
I will stay at the place of my friend. (bad)
Instead, they would say:
I will stay at my friend's place.
You could say this, because possessive pronouns like "his" work better in double possessives:
I will stay at that place of his.
"That" usually sounds better than "the" in double possessives. In fact, "the" would sound awkward in the above sentence.
There's some more information on this page:
http://www.quickanddirtytips.com/education/grammar/double-possessives
