There were four main parts to the Anaconda Plan, which, incidentally, was a very sound military strategy by the North to win the war. The problem was, it would take four years for them to achieve the four parts, when they should have been able to finish it much earlier.
Part 1 called for blockading the South with the Union navy, preventing them from selling cotton to England or getting needed war materials from other countries.
Part 2 called for occupation of the entire length of the Mississippi River, denying the South its only waterway for transportation and splitting the South effectively in two.
Part 3 called for a drive through the Shenandoah Valley into Georgia and to the ocean, splitting the South yet again.
The final part called for the occupation of the capital city of Richmond, and with it, the Confederate government.
I think one very important aspect of the Battle of Gettysburg that wasn't mentioned in the other posts was the fact that once the South lost that battle, they also effectively lost any last chance they had of the British recognizing the Confederacy and entering the war on their side.
A British emissary was with Lee's Army observing - if Lee could have won that battle, in the second year of the war, that far north, then it may have been worth it for the British to intervene. If they had brought in the Royal Navy (as the French had intervened with their navy during the Revolution) it would have lifted the blockade, provided a market for southern cotton, and given them military and financial aid. By losing Gettysburg, the South lost its last, best and only chance to win the war.
Gettysburg is certainly the more famous of the two and gets most of the attention, but I think Vicksburg was the more important as a turning point.
One main reason Gettysburg is more famous is because it did involve a major defeat for Robert E. Lee, but also because most of the professional photographers were closer to Gettysburg than Vicksburg and they arrived on the scene while the burial details were still active. When they finally arrived at Vicksburg the burial details were finished. What's more interesting, photos of dead bodies or photos of opened fields?
The photos of Gettysburg helped to shock the American public and register that battle in the American memory forever.
GETTYSBURG. This battle was considered the turning point in the war for several reasons. First, the Union victory stopped the Southern invasion which was ultimately aimed at Baltimore and then Washington, D. C. Secondly, it was the first time that the Union army had decisively defeated Gen. Robert E. Lee's Army of Northern Virginia. It also cost the Confederates an enormous number of casualties that could never be recovered.
VICKSBURG. The eventual Union victory at Vicksburg opened up the Mississippi River for future Union expansion. It effectively split the Confederacy, isolating the Trans-Mississippi Department west of the river.
ANACONDA PLAN. This idea, conceived by Gen. Winfield Scott during the early days of the war, was meant to create a stranglehold on Southern seaport cities and the eventual capture and control of all port cities along the Mississippi River.
The Anaconda Plan was the plan for defeating the South that was originally put forth by Winfield Scott. It called for a blockade to keep the South from getting supplies from Europe and an advance down the Mississippi to cut the South in two. From there, the Union would start "squeezing" the South.
The Battle of Vicksburg helped with this plan. It removed the last Confederate obstacle to Union domination of the Mississippi River.
Gettysburg stopped Lee's invasion of the North and made it much less likely that any European countries would recognize the CSA and put pressure on the US to end the war.
I think the Anaconda plan is probably the most interesting part of the question, and since other folks have covered the other parts rather well, I wanted to throw a bit about it in here:
Because the war started over economic causes, and the real solution to the war was not to destroy everything and everyone in the south but to make it economically impossible to continue, the Anaconda plan was a particularly interesting manifestation of that goal. It consisted of the blockade of all Southern Ports and then the campaign to take and control the Mississippi, the lifeline (though some felt the Tennessee River more important) of the South economically starting with New Orleans.
As the war progressed, in some ways the plan was pushed to become more active and to involve more active interdiction of trade, etc., rather than just simple blockading.
There is some debate as to whether the plan was even a cohesive plan at the time or simply a collection of disconnected campaigns that later was compiled into a single "plan."