In a single dissociation, a manipulation (lesion, drug) leads to a change in one mental ability or capacity while leaving another intact. So, one might experience loss of ability to name an items one sees, but still be able to name it when touching it.
In a double dissociation, two experimental manipulations each have unique effects on two dependent variables (usually behaviors of interest). So, one manipulation affects one thing but not the other, and the other manipulation affects the second thing, but not the first.
What we all usually think of is the classic double dissociation of speech and language comprehension. Although both processes pertain to use of language, the brain structures that control them work independently. When Broca's area is damaged, patients may still understand language but be unable to speak fluently. They know what they want to say, but are unable to express themselves. Damage to Wernicke's area leads to patients who may still speak fluently, but struggle to comprehend language.
In general, double dissociation is seen as a much more valuable outcome from a study. And, although I have used the typical neuroscience examples here, designs in other areas of cognitive science also can lead to single or double dissociation.