The acceleration is though to the the result of a constant force which is somehow built-in to space - each volume of empty space contains something that results in this force. There are various ideas, mostly to do with some quantum effect, but no-one really has any idea what it is, and so it's given the pretty meaningless name 'dark energy'.
What will the future be? Dark energy seems to stay constant in a given volume of space, so it won't ever grow large enough to rip galaxies, planetary systems, stars, or even people, apart. All these things will be held together by the vastly stronger forces of gravity, electromagnetism, and nuclear forces. But on larger scales the dark energy force will overcome gravity. In many billions of years from now distant galaxy clusters will have accelerated away beyond our ability to see them. The universe will appear much, much smaller.
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A German professor (can't remember his name, sorry) recently published a paper suggesting that the red-shift we interpret as the Universe accelerating, could also be accounted for if atoms had grown in size since the Big Bang. There is no way of disproving this theory at present! I wonder how the Universe could look in time, if this scenario was correct?
That seemed to be being playful with ideas! I would be surprised if this could not be disproved - all kinds of things would surely change. I just can't think of any right now.
The paper Nonny Short is probably talking about can be found here (i don't know if it's been peer reviewed yet).
http://arxiv.org/abs/1303.6878/
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