If by "cardboard" you actually mean thin pasteboard or material for using in paste-ups, or, for that matter, the sort of pages that come in heavy-duty scrapbooks — then your ordinary paper shredder can demolish it, as long as you do it carefully and don't overload it. As always, though, check your manual.
The real thing — heavier, corrugated cardboard — requires a different level of shredder. Corrugated cardboard, in single sheets or in the form of boxes, can be usefully shredded into packing material. There are about four different types of machines that may apply to your situation.
First is the single-sheet non-corrugated jack-of-all-sizes machine. It will take a single sheet (up to the maximum width the feed throat will allow), perforate, pad, and cut it to the correct size for the cutting section, which then slices it up a treat and makes good, light packing material. This class of machine may or may not have a feed table attached; check around.
Next comes the single-sheet corrugated slicer. It will take one sheet at a time, up to the maximum feed width of cardboard of a maximum weight (such as 275lbs) and make packing out of it.
The next level takes two or three corrugated layers at once, up to a defined maximum total thickness and width. This is useful; boxes need only be flattened instead of being torn apart, and generally the machine can handle staples and other slight construction metal pieces.
The top performers produce a trellis-form packing mat in two passes: the first cuts the box to the width required and the second does the slicing and dicing. These are two-slot models.
Note that all these machines are expensive, rather loud (70db or so), and require lots of lubricant and cleaning at regular intervals; some require dedicated 220 volt outlets. Some have a special vacuum cleaner kit of their own.
Manufacturers include: Intimus, Schleicher, Flexible Packaging, etc.