Each atom is at the corner of a cube and 8 cubes meet at each corner. Therefore each atom is shared by 8 cubes.
But I can't imagine it in space, so please tell how I visualize it.
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Surprisingly, this simple question does not seem to have been asked before. http://en.wikipedia.org/wiki/Bravais_lattice shows all the two dimensional Bravais lattices in extended form (but only the unit cell for the three dimensional ones.)
Here is the 2-dimensional image from wikipedia. As you can see in the simple square lattice, each sphere is a the corner of a square and 4 squares meet at each corner. Therefore each sphere is shared by 4 squares.
The simple cubic lattice is the same, except that the atoms are found at the corner of the cubic unit cell. Therefore each atom is shared by 8 unit cells. There are eight eighths (equal to one whole atom) in each unit cell.
I am not aware of any elements that crystallise in the simple cubic lattice. It is not energetically favoured because it is not spatially efficient. The more complex face-centred and body-centred lattices are preferred. Caesium Chloride, Bromide and Iodide form interpenetrating simple cubic lattices with the halide ion at the centre and the caesium ion at the corners of the unit cell (or vice versa depending on your viewpoint.)
Here are the relevant wikipedia links