If you were to break down Zone Blocking into one phrase, it would be "Flank and Anchor".
It was designed so that teams with smaller and lighter (read: ****/cheaper) lineman could still run the ball effectively. The idea was to use speed rather than power to outflank your opponent rather than crash through them.
On the snap of the ball the offensive linemen take a hard step at a 45 degree angle towards where the ball is going, and they try and attack the play side shoulder of the defensive lineman in front of them. They work their hips around so that their bodies are between their man and intended path of the ball carrier. From there, they don't have to drive their man backwards, they themselves just can't be driven backwards until the ball carrier is past them. It's much easier to stalemate than to drive a guy off the ball.
The thing is, that getting to the outside shoulder of a guy that's lined up outside of you is really difficult. A lot of times you won't be able to reach him and he'll shut down the gap the running back is supposed to run through. The beauty of the scheme is that by taking away one gap, he's opening up another gap. It's the job of the running back to see his blockers and understand which gaps are open.
You hear people talk about "One cut runners" and what they mean by that, is that a back gets the ball, sees how his blocks are developing and makes one cut into the open hole. In a power scheme, a running back shouldn't have to read or make "One cut". On a 34 Dive, the 4 hole should be open, and if it's not, the running back isn't at fault. On an Outside Zone Right, there's no hole designated. It's up to the RB to find the hole, because one has to be open unless there's a perfectly called blitz or something like that.
An ideal Zone play looks like this:

The back doesn't have to make a cut. He sees that the DE and the playside LB are both "killed" and the back runs off the TE, untouched into the secondary.
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Sadly for offenses, Zone plays are rarely ideal, just because it's a rare occurence for a TE to be able to seal a DE inside. When the DE beats the TE, the play isn't dead however, if the RB sees it and makes a correct read. So long as the Tackle and the Guard make their blocks, there's a running lane between the Tackle and TE.

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The beautiful thing about the Zone blocking scheme is that if the back has great vision, there's always a running lane. If the Guard gets beat and the TE gets beat, there's still a lane so long as the Tackle "kills" the LB. It's just that instead of being off the ass of the TE, it's now between the Center and the Guard.

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Even if literally every single lineman gets beat, there is still a gap for the back to run through. It's just on the opposite side of the formation.

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Now in real life, most backs won't see any hole farther down the line than the gap between the Center and the playside Guard. We've been cursed with backs with horrible vision. Our RBs are having a good day if they see as far as the gap between the Guard and the Tackle. Arian Foster is a guy who has incredible vision, and you occasionally see him cut across the entire backfield for a big gain.
The farther a play gets strung out, the bigger the holes are on the backside of the play. That sort of freedom allows the lineman on the backside of the play to cut block their men. This is better because cut blocking a guy is much easier than reaching his opposite shoulder. Getting guys on the ground on the front side of a play usually just creates a pile that prevents the back from running, but on the backside taking a DLineman's legs out works because there's space for the back to avoid the pile.
Turn down the volume on your computer if you click on the link, but it should take you to a play against the Cowboys where you see Foster do just what I'm talking about. There are examples throughout the video of what a Zone scheme RB is supposed to look like. He's really good, and you see it because you watch the play and you think he's bottled up and then he makes a cut into space that even from the pressbox view you didn't see.
http://www.youtube.com/watch?v=DgeqUrlmhvg&t=2m20s+++++++++++++++++++++++++++++++++++++++++++++++++