At the beginning of the first episode of the PBS documentary series about the US Civil War the narrator describes some of the main characters without giving their names. One was 'a professor from Maine who executed a text book maneuver saving a Union army and maybe the Union itself' (my recollection). Who was this man and where did this action occur?
The 20th Maine charged down the hill, with the left wing wheeling continually to make the charging line swing like a hinge, thus creating a simultaneous frontal assault and flanking maneuver, capturing 101 of the Confederate soldiers and successfully saving the flank. Source: Wikipedia on J. Chamberlain
The importance of the Union's victory at Gettysburg may be subject to disputes (again, see Wikipedia on those controversies) - but the outcome of that battle is often seen as crucial regarding the further development of the civil war. The defense of the "Little Round Top" for sure wasn't the single decisive moment in the order of events at Gettysburg. But the battle would have developed differently if the confederates had been able to successfully attack the union's troops on their left flank (e.g., on the "Little Round Top").
To build on another answer, the man was Lawrence J. ("Joshua") Chamberlain who was a professor of rhetoric at Bowdoin College in Maine when the Civil War started. He took a leave of absence to join the Union army, determined "to sacrifice the dearest of personal interests to save the country from desolation." Because he was one of very few college men, he rose rapidly to Colonel of the 20th Maine Regiment. After the Civil War, he was elected Governor of Maine.
The "textbook maneuver" he executed was "refusing the flank" (bending back his line in a "reverse L" to prevent it from being overrun by larger number of Confederates). This "set up" the other maneuver of "charging down the hill, with the left wing wheeling continually to make the charging line swing like a hinge," and took place on Little Round Top, a hill on the extreme left of the Union line at Gettysburg on July 2, 1863.
Because Confederate attacks earlier the same day had been repulsed on the right and center of the Union line, this was regarded as the "last chance" for the Confederates to win the Battle of Gettysburg, which in turn was regarded as the last chance for them to win the Civil War.
That's because unlike "Pickett's Charge" the following day, the Confederate attack on Little Round Top had a real chance to succeed, and likewise, the Confederates had a chance to get "compensation" for the loss of Vicksburg and Mississippi to General U.S. Grant (July 4, 1863), until they were driven out of Union territory (Pennsylvania) following the battle of Gettysburg.