1. Supercube is sometimes used as a synonym for hypercube or tesseract, that is to say, a cube extended into the fourth dimension.
2. A (three dimensional) shape generated by replacing the exponent 2 in the equation x2 + y2 + z2 = r2. with any other number.
x2 + y2 + z2 = r2. is the formula for a sphere, but when you replace the exponents with any other number, you will get something like a cube, although it will usually have rounded corners.
If the exponent is changed to a number higher than two, the shape will approach a cube (with rounded corners); the higher the number, the closer to a cube the resulting shape will be. If the exponent is infinity, you will get a perfect cube.
If the exponent is changed to a number lower than two you get various things; 1 will get you an octahedron, 0.5 will get you a cube with concave sides.
There’s a lot more to supercubes, but I am not math-literate enough to understand more than this.
http://www.melbpc.org.au/pcupdate/9605/9605article8.htm
and
http://www.melbpc.org.au/pcupdate/9903/9903article4.htm
Have both pictures and more info.