Why was the Battle of Gettysburg considered a turning point?

There are a number of reasons why the Battle of Gettysburg can be (and usually is) seen as a major turning point in the Civil War.

First, we can say that it was a turning point because it removed any thoughts that President Lincoln might have had about making peace with the Confederacy.  The Union had not necessarily been doing very well in the war at this point, and it was possible that Lincoln would have considered a peace treaty.  The vice president of the CSA was, in fact, making his way to Washington during the battle to discuss prisoner exchanges, but he also had the power to discuss peace.  When Lincoln heard of the victory at Gettysburg, he denied the vice president permission to enter the country.  This showed that he had stopped considering the idea of peace with the South.

Second, the battle boosted morale greatly in the North and lowered it in the South.  Up until this point, General Lee had had an aura of invincibility.  He had seemed to win every battle he fought against the Federal army.  In this battle, however, his forces were defeated in an even fight.  This made it clear that he was not invincible.  In the process, it made Northerners much more confident in their eventual victory and made the South feel much less optimistic.

Finally, and perhaps most importantly, the battle ended the South’s hopes of winning the support of European countries.  The battle happened because Lee had invaded the North.  If he had succeeded, many European countries might have recognized the Confederacy, just as France started to support the colonies after the Battle of Saratoga in the Revolutionary War.  Worse yet, some of them might have even given the South military support.  Either of these scenarios could have forced the North to make peace.

For all of these reasons, the Battle of Gettysburg is rightly seen as a major turning point in the American Civil War.