The Coriolis effect occurs because of Earth's rotation and the fact that the atmosphere and oceans are not "connected" to the solid part of the planet.
1. Atmospheric Circulation Patterns
Earth rotates eastward. How do we know? Because the Sun rises in the east. Visualize it using a globe.
If Earth rotates eastward, and the atmosphere is not connected to the solid part of Earth, then which way must the atmosphere flow? Imagine that you are in a car at a stoplight. When the light turns green and the car goes forward, which way do you go? That's right... you go backwards, in the opposite direction, because you are not connected to the car (that's why you need to wear a seatbelt). So, if we apply the same thinking to the Earth and its atmosphere... if Earth rotates eastward, then the atmosphere, at the equator, must flow in the opposite direction, westward.
Between 0 and 30 degrees North and South latitude, the atmosphere flows westward. Between 30 and 60 degrees North and South, the atmosphere flows eastward. And, between 60 and 90 degrees North and South, the atmosphere flows westward again.
If you live in the United States, between 30 and 60 degrees North latitude, then you will notice that weather moves across the country from west to east (that is, eastward). Why? Because of the prevailing westerly winds, which push weather systems across the U.S. from west to east. By the way, winds are named for the direction from which they are coming... so westerlies come from the west and move towards the east.
2. Oceanic Circulation Patterns
The winds drive the oceans, so you will notice that oceanic and atmospheric circulation patterns are very similar. The result is that in the Northern Hemisphere, oceans flow clockwise in their basins (think North Pacific and North Atlantic Oceans). In the Southern Hemisphere, they flow counter-clockwise.
3. Flight Paths
Anything that flies (planes, birds, missiles, space rockets) is affected by the Coriolis effect. For example, a plane that flies along a North-South path must not fly directly towards the target location. Instead, it must fly either slightly right or left of it (depending on which hemisphere it is in) in order to hit the target location.