When black-boxes are found, how is the information retrieved from within the data storage? Is there a special kind of device used for this?
They send it to the manufacturer (or other lab) who knows how to get the data out.
Old black boxes stored the data on tape which they take out put in a reader and print out the data. Partially damaged tape can still be read, the result will be a data stream with holes in it.
Newer black boxes use solid state storage (like one of those 8gig memory cards). It will be put in a reader to read it out. The exact format of the data will be specific per manufacturer.
Then the analysts will make nice graphs of it for the investigators.
If the blackbox is intact, procedures described in other answers would apply. If, however, the data storage device (ssd, tape, etc.) itself is damaged, it will be given to forensic experts such as police criminal unit or other expert group or company that specializes in data recovery. For them it doesn't make much difference that the storage device is in fact an aircraft flight data recorder storage, they handle it in pretty much the same manner as any other device.
Blackbox itself is unimportant, only the data on storage devices is.