Grammatical name and function of the following body [closed]

'At the end of the forty days of mourning...' – what is the grammatical name and function of this expression?

This type of construction is called nested or embedded prepositional phrases. In your example, the first prepositional object end is modified by a prepositional phrase whose object is modified by yet another, i.e., one prepositional phrase inside another inside yet another, thus “nested”:

{ at end [ of days ( of mourning ) ] }

There is a limit to how long readers will hold their mental breath as they move through such nested constructions. Three seems a reasonable limit:

The police found the revolver behind a clock on the south kitchen wall of his small apartment in Brooklyn.

Four nested prepositional phrases is over the limit for most readers, but three still sounds reasonable:

The police found the revolver behind a clock on the south kitchen wall of his small Brooklyn apartment.

So a good rule of thumb is “one two, three, too many.”