The Fill Style dropdown is used to define fill styles for objects based on polygons or arcs.
Transparent and Open are similar in that they do not make the filled object opaque. The difference is that the Transparent fill style is used for closed objects (e.g. rectangles, polygons and beziers whose start and endpoints are the same). If you have drawn a polygon whose startpoint is different from its endpoint and you apply a Transparent fill, a line will be drawn to connect the start- and endpoints. This also applies to beziers and arcs - the Transparent fill closes the object. The Open fill style means that a many-sided object or arc is left open and no additional line is drawn.
A Solid fill style fills the selected objects with the color set in the Fill Color box.
Hatch patterns (e.g. DiagonalCross, Vertical) fill the selected objects with a black pattern regardless of the Fill Color.