Like PP, I think it's probably juice-related. Most apple juice comes from dessert apples and has lots of tartness (mostly malic acid) to balance the sweetness. Ferment out the sweetness, and all that's left is...
Have your previous batches fermented completely dry? (Usually the case with cider)
Did you use the same yeast? (Which one?) If you didn't pitch yeast, and relied on the wild yeast, there's your answer. Wild yeast is known for unpredictability.
Is it in fact "tartness" - the perception of malic acidity - or is it sourness, like a little vinegar got in? That would be an indication of oxidation; cider is extremely susceptible to aeration after fermentation. It can be a little tricky differentiating between malic acidity and acetic (vinegar) acidity.
Or, maybe, it's a different sort of flavor altogether; somewhere between sour and bitter, a kind of funky effect that is hard to pin down. I get that a lot in my ciders, and it's probably Brettanomyces, and in the proper proportions it is actually a benefit - but too much is a defect. Brett keeps working long after the yeast quits, so this flavor often develops after fermentation appears to be finished.
As to fixing it, the best suggestion I can offer is to add honey. One or two cups in a 5 gallon batch can make a real difference, not just the sugar but the flavor can improve a marginal cider. Of course it is likely to re-ferment, so be careful about bottling.. if it's in a keg you should be OK.
If it's acetic (vinegary) there is no cure, and your best bet is to aerate the heck out of it, add some oak chips, and turn it into cider vinegar. Which can be great stuff, so it's not a total loss.