The Black and White Dots illusion is an enhanced version of the Hermann Grid illusion, originally described by the German physiologist Ludimar Hermann in 1870.
In the standard Hermann Grid Illusion, grey spots are seen at the intersections of the white lines, even though there is no actual shading there.
The enhanced 'dots' variation, discovered by E. and B. Lingelbach & M. Schrauf in 1994, superimposes white circles on a grid pattern having grey lines. This produces a 'scintillating' effect in which the white circles in the periphery of vision appear black when the eyes move across the image.
When the eyes are perfectly still and focus on one of the white circles, the black dots appear grey (or may disappear) while the focussed-upon white circle is seen as white.
For more information and theories about the grid illusions, see this Wikipedia article.