The Earth rotates about it's axis once every 24 hours. All objects on the surface of the Earth (including us) are carried along with it, and are constantly moving around in a large circle in space. The Earth's oceans and atmosphere are also spinning around with the Earth at the same rate. The speed of movement is highest at the equator, and decreases gradually both north and south, until it reaches zero at the two poles. Therefore, the Coriolis Effect is greatest near the equator, and decreases towards the poles. [Note that some maps show much greater distortion of trajectories at the poles than at the equator, but this is because 2D maps are greatly distorted at extreme latitudes due to the map's projection. This has nothing to do with the Coriolis Effect.]
Over short distances, this effect is negligible, and we do not usually consider it. Slower moving objects, just as pedestrians, cars, or ships at sea can also ignore it. But objects that either travel at high speeds or remain airborne for a long time need to consider the Coriolis effect, or they will go off track.
The Coriolis Effect has been understood for a long time. The equations were published in 1835, but even prior to that balloonists knew that simply by taking off and staying put, you could land a long distance away because the Earth would rotate under you while you were aloft, which is an example of the Coriolis Effect. On much smaller time scales, a famous example is the shelling of Paris during the First World War by the Germans. The Germans built huge rail-mounted cannons, called the "Long Max" cannons, which had a range of 75 miles. Firing from the northwest of Paris, the Germans kept missing Paris and the shells would land west of Paris, because during the long flight of the shell, Paris had moved eastwards. They had to make corrections for the Coriolis Effect before they could hit their targets.
Modern aircraft on long distance flights also have to compensate for the Coriolis Effect. In other words, they aim for where they expect the landing site to be, at the time they reach their destination. If they are traveling north in the northern hemisphere or south in the southern hemisphere (away from the equator), their path is deflected eastwards. If they are traveling south in the northern hemisphere or north in the southern hemisphere (towards the equator), their path is deflected westwards.
The basic principle is simple. If you are in a fluid medium (meaning the atmosphere or the oceans) where lateral slippage is possible, and if you then move north or south (but not east or west), then your path will appear to be deflected east or west, to an observer on the Earth. An observer in space will not see this deflection - instead, he will see the Earth rotate under you, so the net result is the same - that you come to land on a spot east or west of where you had been aiming when you took off.
The graph on the right shows this deflection when traveling north in the northern hemisphere. Since you are moving away from the equator, the deflection is eastwards.
Think of it as two vectors, one pointed to the right, representing the Earth's rotation (and yours before you take off), and the other representing your flight under power to the north. Since these two vectors are orthogonal to each other, neither affects the other. Your actual path should be a straight line north-east.
But as the graph shows, the path is not a straight line but rather a curve. This is because the Coriolis effect is not constant across the earth, it changes continuously as you move away from the equator.
Sometimes the Coriolis Effect is referred to as the "Coriolis Force". This is not really a force. We can consider it a pseudo force in the sense that it accounts for the observed actual motion. However, the more fundamental reason for the motion is not that there is any force acting on the object to make it follow that path, it is simply that the Earth itself is rotating during the course of the movement (the flight), and therefore the end point doesn't correspond to the direction aimed at.
The Coriolis Effect is very weak. Some people have argued that drains in the northern and southern hemispheres twist water clockwise or anticlockwise due to the Coriolis Effect. This is incorrect. While the water swirling in a drain is indeed subject to the Coriolis Effect, this effect is tens of thousands of times weaker than other factors such as turbulence, and is completely overpowered by them. The Coriolis Effect is only relevant when we are talking about either high speeds and/or very long transit times.
The Coriolis Effect is very important for weather systems. The eastward and westward deflection of the trade winds and anti trade winds is an example of the Coriolis Effect.
The trade winds are permanent winds which flow between the equator and the Tropic of Cancer in the northern hemisphere, and between the equator and the Tropic of Capricorn in the southern hemisphere. They were called "trade" winds because they were used by navigators during the days of sailing ships for trading voyages to India and the far east. The westerlies, or anti-trade winds are permanent winds between the Tropic of Cancer and the Arctic Circle in the northern hemisphere, and the Tropic of Capricorn and the Antarctic Circle in the southern hemisphere. As you can see in the diagram above, all these winds are deflected by the Coriolis Effect.
The continental US lies in the northern anti-trade wind zone. As you can see in the diagram above, the prevalent winds are therefore from the southwest towards the northeast. If you watch the weather news on TV often, you may have noticed that weather systems across the US tend to move from the Pacific coast towards the Atlantic coast, in a northeasterly direction. This is because of the anti-trade winds, which blow across the US from southwest to northeast, driving weather systems in that direction.
North of the Arctic Circle and south of the Antarctic circle there are polar winds, which blow from the poles towards the equator. Their direction is also modified by the Coriolis effect as shown.
The diagram above shows the movement of air between the upper and lower atmospheres. The winds we feel are due to movements of air at the surface. However, for obvious reasons, if a lot of air moves from some region A to another region B, then there must be a reciprocal path for air to move away from B, since air cannot accumulate at any given region indefinitely. This reciprocal circulation is provided by the upper atmosphere circulation, which happens near the edge of the troposphere, 7 to 17 kilometers above the surface of the Earth. The Jet Stream is part of the upper atmospheric circulation.
Temperatures are very high near the equator (shown by the red area near the equator in the diagram above), which causes the air there to expand and lose density. This heated, expanded air rises up into the atmosphere, further lowering the pressure at the surface. This causes air to rush in from high pressure areas, due to the pressure differential. The high pressure areas near the equator are the two tropics - at latitudes of 23.5 N and 23.5 S. Air rushing in towards the equator from the tropics produces the trade winds.
As the air at the equator rises, it joins the upper atmospheric circulation, and is carried away. It moves towards the tropics in the upper atmosphere. Recall in the previous paragraph we mentioned that at the Earth's surface, air moves from the tropics towards the equator. This movement of air away from the tropics at the surface, creates a low pressure area at the tropics in the upper atmosphere, since air in the upper atmosphere sinks down at the tropics to compensate for the air moving away from the tropics at the surface. Because of this upper atmospheric low at the tropics, upper atmospheric circulation from the equator is sucked towards these lows. This forms a complete, closed cycle - up at the equator, north or south to the tropics through the upper atmosphere, sinks back down to the surface at the tropics, then moves back to the equator as surface winds. A closed cycle like this is called a Hadley Cell.
There are also high pressure areas on the two poles, because of the extreme cold. Similar cycles are set up between the poles and the arctic and antarctic circles.
Vertical air speeds over different parts of the Earth, July averages. Shows areas where air is rising up from the ground into the upper atmosphere in shades of red-yellow, and areas where the air is descending from the upper atmosphere to the surface in shades of blue. From Wikipedia.
While these pictures make things look very neat and pretty, the real situation is not so neat. The description above is represents an ideal situation, perhaps what the Earth would look like if it were covered by one uniforms ocean of water, with even heating/cooling characteristics. However, the Earth has sizeable landmasses, and most of these are concentrated in the northern hemisphere. Further, the land masses are not uniform, but contain a mix of features, such as flat plains, mountain ranges, lakes, high plateaus and low lying depressions, etc. This greatly modifies the simple theoretical situation described above. You can read more about how altitude affects the climate in the section here.
It's also important to remember that these zones shift seasonally between summer and winter, as the area of the Earth under the Sun's direct light shifts northwards and southwards. This shift is on the order of about 28 km per day, so it is quite significant over even a few days or weeks.
The figure to the right shows actual high and low pressure areas on the Earth, as an average for the month of July. The colors and numbers represent vertical air speeds -- air rising from the surface into the upper atmosphere, and air descending from the upper atmosphere to the surface. Red areas show rising air (corresponding to patches where air gets heated and rises), while blue areas show falling air (relatively cooler spots where air descends to the surface).
As you can see, land masses break up the pattern into a number of separate cells (called Hadley Cells, mentioned below). Most of the hot patches lie over the oceans -- there is a whole band across the equator, as well as large patches in the Atlantic and Pacific. Next to it is a band representing the tropics, where air sinks back to the surface. This band is very clearly represented in the northern hemisphere, as a thin blue band running across the map near the latitude of Central America. However, its counterpart in the southern hemisphere is not so clearly seen. In the southern hemisphere, the equatorial red band seems to extend far south. There are two reasons for this - first, because there is not much land in the southern hemisphere at these latitudes (you can see the blue reappear where there is land, such as off the mountains on the west coast of South America), and oceans heat up or cool down much more uniformly than land. Second, because July is winter for the southern hemisphere, and circulation patterns are different from what they would be in the summer.
You may have noted that the high and low pressure areas are located at very specific places - at the equator, the poles, the two tropics (23.5 N and 23.5 S), and at the arctic and antarctic circles (66.6 N and 66.6 S). What is so special about these places, why should high or low pressure areas form there?
The equator and the poles are easy to understand, since they are the hottest and coldest spots on Earth, in terms of latitude. But what makes the tropics special? The tropics are important because the Earth's tilt is 23.5 degrees, so the region between the two tropics forms a band in which it's possible to have the Sun directly overhead during some part of the year. If you go past the tropics, the Sun will never be directly overhead, it will always be a bit south (if you're in the northern hemisphere) or a bit north (if you're in the southern hemisphere). So the tropics mark the boundary of the band which receives the maximum direct sunlight from the Sun. Past this band, sunlight is slanted and therefore weaker.
The arctic and antarctic circles are special for a different reason. They mark the uppermost and lowermost extremes from which the Sun can be seen year round. If you go north of the arctic circle, then there will be some days in the year when the Sun never rises above the horizon. The same thing is true if you go south of the antarctic circle. So they mark the boundary between zones which have day night cycles throughout the year, and zones which have some days of perpetual daylight during the summer and some days of perpetual night during the winter. The closer you get to the poles, the more such periods of perpetual day or perpetual night there will be each year. At the extremes, if you are on either the north or south pole, the Sun will not sink below the horizon for 6 months of the year, and it will not rise above the horizon for the other 6 months.
As you can see, the tropics as well the arctic/antarctic circles demarcate natural boundaries, which are the direct result of the Earth's axial tilt. This is why, over millions or billions of years, weather patterns have stabilized at these natural boundaries.