Double Dissociation is when two related mental processes are shown to function independently of each other.
A classic example of Double Dissociation is speech and language comprehension. Although both processes pertain to use of language, the brain structures that control them work independently. When a part of the brain called the 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. On the other hand, when a part of the brain called Wernicke's area is damaged, patients may still speak fluently, but be unable to comprehend language. This results in properly constructed but nonsensical sentences.
By establishing Double Dissociation, scientists are able to determine which mental processes are specialized to certain areas of the brain.