An inscribed, or cyclic, quadrilateral is one where all the four vertices lie on a common circle. Another way to say it is that the quadrilateral is 'inscribed' in the circle. Here, inscribed means to 'draw inside'.
In the figure above, as you drag any of the vertices around the circle the quadrilateral will change. Note that if you drag a vertex past an adjacent one, the quadrilateral will be 'crossed'. It will have one side that crosses over another. As with all polygons, this is not regarded as a valid quadrilateral, and most theorems and properties described below do not hold for them.
In a cyclic quadrilateral, opposite pairs of interior angles are always supplementary - that is, they always add to 180°. For more on this see Interior angles of inscribed quadrilaterals.
If you know the four sides lengths, you can calculate the area of an inscribed quadrilateral using a formula very similar to Heron's Formula. For more see Area of an inscribed quadrilateral.
It turns out there is a relationship between the side lengths and the diagonals of a cyclic quadrilateral. For more see Diagonals of an inscribed quadrilateral.