An anticyclonic storm is a weather storm where winds around the storm flow contrary to the direction dictated by the Coriolis effect about a region of low pressure. In the northern hemisphere, anticyclonic storms involve clockwise wind flow; in the southern hemisphere, they involve anticlockwise (also called counterclockwise) wind flow.
Anticyclonic storms usually form around high-pressure systems. These do not "contradict" the Coriolis effect; it predicts such anticyclonic flow about high-pressure regions. Anticyclonic storms, as high-pressure systems, usually accompany cold weather and are frequently a factor in large snowstorms.
Anticyclonic tornados often occur; while tornados' vortices are low-pressure regions, this occurs because tornados occur on a scale small enough that the Coriolis effect is negligible.