Soon after sunset, as the sky gets dark, look toward the south and you'll immediately notice the brilliant planet Jupiter, but there's another night sky target that also promises a great experience: the amazing Andromeda Galaxy.

Jupiter is a great starting point to find the Andromeda Galaxy. After you find the bright planet, look high above it – almost directly overhead - to find four bright stars. These are the Great Square of Pegasus, the Winged Horse, an unmistakable star pattern, even though it is slightly battered out of true square shape. 
Nonetheless, Pegasus is a striking figure and once you see it you won't forget it.
Interestingly, the star in the upper left corner of the Square – Alpheratz – actually belongs officially to the constellation Andromeda. Andromeda, according to legend, should be chained to a rock.  Instead she seems to have become chained to the horse: a double strand of stars – one strand bright, the other dim – connected to the upper left corner of the square. 
About midway and above these star strands is the Andromeda Galaxy, one of the most amazing and fascinating of sky objects. It can be seen faintly with the unaided eye, but binoculars and small telescopes promise better viewing.
This sky map shows where to look to see the Andromeda Galaxy over the next week, though clear weather is vital to spot it.
In the year 905 A.D., the Persian astronomer Al Sufi drew attention amidst the stars of the Andromeda constellation to a “Little Cloud,” which appeared on star charts long before the telescope was invented in 1609. 
If the sky is clear and moonless you can indeed see an elongated hazy patch with your unaided eye about as long as the width of the full moon and half as wide.
Through binoculars and telescopes it remains an elongated patch which gradually brightens in the center to a star-like nucleus. It was listed as object number 31 in Charles Messier's 18th century catalogue of galaxies, nebulas and star clusters, which is why it is known as Messier 31 or M31. We know it better as the Andromeda Galaxy. 
The best way to positively find the Andromeda Galaxy is to focus your eyes or binoculars on Alpheratz. Run straight across to the left and get the star Mirach in your field of view. 
Then run slowly upwards to a fairly bright star above Mirach and continue to run up in roughly the same direction and same distance. You'll immediately take note of a little patch of faint light.  
Please forgive this patch of light for being so faint and tired looking. It's amazing to realize that, as you see it tonight, this light has been traveling some 2.5 million years to reach you (give or take a few hundred thousand years), traveling all that time at the tremendous velocity of about 671 million mph – the speed of light.
The light from that “Little Cloud” is actually the total accumulation of light from over 400 billion stars.
As you look at the Andromeda Galaxy tonight you'll be doing something that no one else in the world except a stargazer can do: You will actually be looking back into the distant past. The light you are seeing is around 25,000 centuries old and began its journey around the time of the dawn of human consciousness. 
(Source: Space.com)