A square inscribed in a circle is one where all the four vertices lie on a common circle. Another way to say it is that the square is 'inscribed' in the circle. Here, inscribed means to 'draw inside'.
The diagonals of a square inscribed in a circle intersect at the center of the circle. To see this check the 'diagonals' box in the applet above. As with all squares, the diagonals bisect each other at right angles.
Another way to think of this is that every square has a circumcircle - a circle that passes through every vertex. In fact every regular polygon has a circumcircle, and so can be inscribed in that circle.
You can construct a square inscribed in a circle using compasses and a straightedge. For more see Constructing an inscribed square.