As you may have learned, the Coriolis effect arises from the earth's rotation, and the fact that objects near the equator are spinning faster than those near the poles. The Coriolis force affects the path of things that are traveling relative to earth's surface: for example, air currents.

Weather systems are regions of air at different temperatures that move around the globe. The movement of air (hot air rising, for example) causes changes in local air pressure. Storms like hurricanes and cyclones are examples of low-pressure systems; these will suck in air towards the center of the storm. The currents of air traveling towards the storm will be influenced by the Coriolis force, and their trajectories will be curved. This is why storms rotate. Due to the Coriolis force, storms in the Northern hemisphere rotate counter-clockwise, whereas in the Southern hemisphere, they rotate clockwise.

The strength of the Coriolis force depends on the speed of travel and the speed of the earth's rotations, and its effects are more pronounced over very long distances. It is too weak to control the direction of water going down a sink drain- because that water is moving quite slowly and traveling only a short distance. Air currents and weather systems traverse the globe and move at high speeds, so they are strongly affected. It is also interesting to note that other planets rotate a lot faster than earth- Jupiter, for instance- and experience larger Coriolis forces as a result.

Here is a link with a short explanation of the Coriolis effect and more discussion of how it influences weather patterns: