You can only cut along edges, and the shape must be in one piece and flat. By unique, I mean rotations and reflections don't count.

This problem can be analogized to four dimensions as well. How many unique 3-dimensional shapes can be made by "unfolding" a 4-dimensional hypercube into 8 cubes? This problem is significantly more difficult than the first.

Most people find the concept hard to grasp.  The best way to understand is to use analogies.  A line segment is to a square as a square is to a cube as a cube is to a 4-d hypercube (also known as a tesseract).

Take a square.  One way to look at it is as 2 parallel line segments, with the ends connected by 2 more congruent line segments.
Take a cube.  One way to look at it is as two parallel squares, with the corners connected by line segments congruent to the sides of the squares
So a tesseract would be two parallel cubes with all the vertices connectedby line segments congruent to the sides of the cubes.

Another way of looking at a cube is this:

________
|\    /|  This is a central projection.
| \__/ |  The smaller square inside is actually
| |  | |  the same size as the larger one,
| |__| |  but it apears smaller because it is
| /  \ |  further away.  Notice the shape of the
|/____\|  six square faces.  They are distorted.

Similarly, a hypercube can be seen as a small cube inside a bigger one, with all the vertices connecting.  It has 8 "faces," but the faces are 3-d cubes, not squares.  They, too will seem distorted because we are flattening the tesseract into 3 dimensions.
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I can't find a better way to explain the unfolding of a tesseract than this animation.