I'm surprised why people call a Rubik's cube as $2 \times2\times2$ or $3\times3\times3$? It's a CUBE! Isn't saying $2\times2$ or $3\times3$ cube (or rather just $2$ or $3$) sufficient?
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closed as off topic by Clive Newstead, arjafi♦, Hagen von Eitzen, Michael Albanese, Davide Giraudo Feb 3 '13 at 12:24
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Cube is a 3D object. It has got three dimensions. So basically, $3 \times 3 \times 3$ means 3 columns, each column with 3 boxes and 3 rows of such columns .
Or in terms of mensuration, you can think of it as a cube 3 units long, 3 units wide and 3 units high.