This puzzle is devious. Externally, it looks like a cube with dovetails cut on four sides—which should be impossible. Dovetail joints are normally cut to join two planes to each other, like the front of a drawer with its side. It shouldn't be possible to cut dovetails across four interlocking planes...so how does this puzzle do it?
You can just skip ahead to the video below if you'd like a visual explanation—or I'll spoil it for you now. In short, the "impossible dovetail" relies on a hidden set of diagonal cuts that allow the puzzle to slide apart diagonally. There's often a pin or other fastening mechanism to prevent an accidental slide, making the joints appear firm and fast until you press it.
The logic of this puzzle plays on our assumptions about how dovetail joints work. Because these joints typically prevent motion in two planes, those are the planes we look at. Is it possible to pull the top off the puzzle? Nope! How about side-to-side? Impossible! By adding a dimension to the puzzle, the solution is revealed in that extra dimension as well—you've got to slide the pieces apart diagonally. The joints, of course, aren't impossible...it's our assumptions about how dovetail joints are cut that are incorrect.
In this video, "Mr. Puzzle" walks us through the solution to a wooden version of the puzzle (the solution starts at 1:52). Have a look:
If you're curious how this thing was made, here's a similar project by Clickspring, but machined out of aluminum. Behold: