Tl;dr: The three-judge appeals court panel affirmed a summary judgment of invalidity. The panel announced a pretty straightforward test as one "filter" that can rule out patentability for certain claims:
1. If a method can be performed by the human mind alone, or with paper and pencil, then it's an "abstract idea," and therefore unpatentable under Supreme Court precedent.
2. If as a practical matter [a] the method cannot be performed with paper and pencil [b], then it passes the filter described in 1 above [c].
3. If a method would be unpatentable under 1 above, then a claim to a storage device encoding computer instructions for performing the method is likewise unpatentable.
NOTES:
[a] In its discussion, the court distinguished a couple of prior cases where such non-paper-and-pencil methods had been held patentable. See pages 20-21 of the PDF. EDIT to respond to petegrif's comment below: One of the earlier methods that had been ruled patentable was "for rendering a halftone image of a digital image by comparing, pixel by pixel, the digital image against a blue noise mask ...." The CyberSource panel distinguished this precedent by saying that, as a practical matter, that earlier method could not be performed by the human mind or with paper and pencil; see page 21 of the PDF.
[b] An unanswered question: How exactly are ordinary people supposed to determine whether a method can or can't be performed with paper and pencil "as a practical matter"?
[c] Even if a method gets past the "paper and pencil" filter, there are still other filters to pass before it will be patentable---but once the Patent and Trademark Office has issued the patent, then a presumption of patentability applies.
COMMENT: This is an area where the net effect of the precedents is not entirely clear. It wouldn't surprise me if the court were to take this case en banc, namely to have all of the court's 12 judges rehear and redecide the case in an attempt to clarify the law. (An en banc opinion is generally regarded as carrying more precedential weight than an opinion by a three-judge panel.)