What is the significance of the setting of A Midsummer Night's Dream? What major shifts in locale take place, and when do they occur?

Shakespeare uses two distinctly different locations in the
play A Midsummer Night's Dream.  One (the setting at the beginning
and end of the play) is the court and city of Athens in ancient Greece, and the second
is an unnamed forest, or woods (the setting of the middle section of the
play).

There are a few significant observations to make
about these two settings:

1. The first setting,
   Athens, represents the "real" world with all its problems and mundane human daily life. 
   The second setting, the woods outside Athens, represents the fantastical events of a
   dream.  At the end of Act I, when the setting switches to the forest, it signals the
   beginning of the "dream," and when the Lovers and Bottom are awakened in Act IV, they
   are literally returned to their real lives in the mundane world of
   Athens.


2. The world of Athens and its court is one of law
   and order, ruled by the logic of the laws, rather than by emotion or feeling.  This is
   displayed by Theseus as he sides with Egeus over Hermia's impassioned plea.  He must
   choose the law.  In the forest, however, Titania and Oberon rule with their emotions,
   plotting and scheming against each other, but just as swiftly overturning and forgiving
   when their moods have shifted.  So, the world of Athens is one of law and order, and the
   world of the Fairies in the forest is one of emotion and
   caprice.


3. For Elizabethans, the woods represented magic
   and mystery and fairies really did live there.  It was a magical and somewhat forbidding
   place, where unexplained and sometimes frightening events could transpire. 
   Thematically, this relates to a darker, hidden world, while the world of Athens is a
   light and predictable (though sometimes unhappy)
   place.

To answer your specific question about
the shifts, they happen between Acts I and II, when the scene changes from Athens to the
woods, and then, during Acts IV and V, there is a more gradual transition back to
Athens.

Theseus, Hippolyta and Egeus wake the Lovers as
they are hunting in Act IV, thus bringing a bit of the law and order of Athens to the
edge of the woods where the Lovers sleep.  Theseus, it seems, is affected somewhat by
his proximity to the world ruled by the emotions (the woods), as he decides to pardon
Hermia for her disobedience and allow the Lovers to join himself and Hippolyta on their
wedding day.

Act IV, scene ii brings the play fully back to
the daily world of Athens, as the Mechanicals, gathered at Quince's house, await the
return of Bottom.  As a note:  Quince's house is also a "setting," but is contained
within the perimeters of the world of Athens, so I have not listed it as a "major"
location.

For more on the settings of A Midsummer
Night's Dream, please follow the links below.