As well as being arguably the most important battle in the history of the United States, Gettysburg is also one of the most written-about. You can find detailed narratives of every hill taken and every bullet fired elsewhere on the internet and I won't repeat them here. The point of what follows is to explain why what happened on the field at Gettysburg mattered to the course of the American Civil War and the entire future of the Republic; in so doing, it is a potted analysis of the civil war itself.
First, a few facts. The Battle of Gettysburg was what is called a "meeting engagement", when two opposing armies accidentally run into each other and then end up having an enormous battle. Neither the Union or the Confederacy planned to fight at Gettysburg. The town had been garrisoned by a small Union force, which was stumbled over by a small Confederate force which had heard about a much-vaunted supply of shoes in the town and had been sent to "get those shoes", footwear not being quite as ubiquitous in the under-supplied Confederate forces as the men would have liked. From there, it steadily escalated.
Gettysburg is in Pennsylvania, which meant the Confederates were quite deep into Union territory. Not long before, the rebels had inflicted fairly major defeats on the Union at Fredericksburg and Chancellorsville. But it wasn't enough. It was 1863, and even this far into the rebellion the South saw no clear way to victory. They kept beating the North in battles around Virginia but they knew that in the long-term, the awesome industrial and manpower resources of the Union would prevail if they kept fighting a war of attrition. The Confederacy would run out of men before the Union did, and President Jefferson Davis and General Robert E. Lee knew it.
The solution that Lee and Davis decided on was to move beyond a war of attrition. Lee decided to sally his forces into Union territory, threaten key cities and possibly inflict a humiliating defeat on the Union, and then reap the political and diplomatic benefits. The inability of the Union to keep Lee's forces off their home turf might persuade overseas powers like England and France to recognize the Confederacy as a separate state, which would boost their chances of never been forcibly reunited with the North; similarly, it might distress and depress northern public opinion and lead to Abraham Lincoln being defeated in the 1864 election or having to bend to the wishes of those in the North who wanted to cut their losses and let the South go. It might even save slavery. Lee's thrust north was also driven by the simple fact that for years, his army had been steadily stripping the Virginian countryside clean of food and forage until nothing was left; by giving Northern civilians a taste of what it was like to be occupied, he might persuade them to turn against the war.
The need for a stunning victory was what led the rebels to make what could be seen as a strange strategic choice. The Confederacy was essentially an insurgency, and insurgencies can often win just by not losing. A low-grade guerrilla war that burned slowly for a decade, or a defensive strategy which dared the Union to keep smashing up against a wall of steel in Virginia might have eventually convinced war-weary northern public opinion to let the southern states go. Instead, Lee - flush with previous victories and knowing that a recent conscription act in the South meant his manpower was now at its peak possible level - instead went on the offensive.
This was risky because technology at the time greatly favoured an army which was playing defence. The American Civil War has rightly been called a forerunner of World War I in this regard, and although the meat-grinder churned through both sides the one on the defensive usually came off better. At Gettysburg, the Union ended up dug in on the high ground (Cemetery Hill) with the rebels trying to assault them. Lee risked everything in this final assault and he lost, as some of his key subordinates warned him he would. Although both sides lost about a third of their men, this was much harder for the Confederacy to bear due to its smaller population. Lee had carried out a daring maneuver into enemy territory and then ended it with a dumb assault in which he forfeited his natural advantages. He was back to trying to attrite the Union forces, but this time he came off worse.
Gettysburg wasn't exactly a decisive victory for the Union, either - Lee's army slinked back to Richmond, while Lincoln was incensed by the fact that General George Meade did not pursue the Confederates and deal a killer blow. Meade was accused of being a traitor and faced Congressional investigations, making it clear that contemporaries did not recognize that Gettysburg would prove to be the "high water mark of the Confederacy", as it is often called.
But the battle marked the point at which the initiative clearly moved to the Union in the war, and when all hopes of diplomatic recognition for the Confederacy by outside powers vanished. It consolidated Lincoln's position and moved him to some of the greatest rhetorical heights in American history in the Gettysburg Address and his Second Inaugural Address; this was not insignificant as it symbolized the newfound determination of the Union to prevail and subdue the South at all costs. When two democracies are at war, public sentiment matters a great deal, and it was after Gettysburg that the blues began to become sure they would win and the greys sure they would lose. The Confederate strategy of seeking to inflict a decisive defeat on the Union had failed, and after that had failed they became as strategically bankrupt as they were morally.