In this post I will explain Exposure Compensation in very simple terms with the goal to focus on the practical application of this feature, and not at providing a precise technical explanation of what it is;
In the past setting the exposure that resulted in a properly exposed picture required at least some experience from the photographer. Today's digital cameras contain computer chips storing information that allows the camera to choose and set a proper exposure for most shooting situations. This stored information is almost like an expert photographer that is always available to help.
No matter how much information is stored in the camera and how advanced the camera exposure system is the "perfect exposure" is a matter of personal opinion. Many cameras have a feature that helps to fine tune the already acceptable exposure. They allow the photographer to change the exposure in discrete steps according to his/her personal preference. The feature is called "Exposure compensation" (abbreviated here as "EC"). Simply put EC is the ability to make the picture slightly darker or slightly lighter:
(1) Exposure compensation -1 step
(2) Exposure set by the camera
(3) Exposure compensation +1 step
Most cameras allow adjustments of the EC 2 aperture steps up or down in 1/3 step increments. What you will most likely see on your digital camera looks like this: -2.0, -1.7,-, -1.3, -1.0, -0.7, -0.3, 0.0, +0.3, +0.7, +1.0, +1.3, +1.7, +2.0.
You can darken or lighten a picture using the shutter speed or the aperture. For example you have an image which according to your camera is correctly exposed with an aperture of 4 as shown on fig. "f4" below. To darken it you would close the aperture from 4 to 5.6 and retake the picture. The result is shown in fig. "f5.6". The image is darker but it may be too dark for your taste. The exposure that you are seeking maybe somewhere between an aperture of 4 and 5.6. Your camera may not have such a setting.
In this case you would use EC. The three pictures under "f4" and "f5.6" show the fine variations possible with the use of EC - "f4 and EC -0.3", "f4 and EC -0.7", and "f4 and EC -1.0".
When setting the aperture to the original 4 and using EC of -0.3 (1/3 of a full aperture stop) you are in effect setting the aperture to about 4.5. A darker picture is taken with EC set to -0.7. And of course a darker one with EC of -1.0.
If you compare fig. "f5.6" and fig."f4 and EC -1.0" you will notice that they are very similar. That is so because using aperture 4 with EC of -1.0 will produce the same result as using an aperture of 5.6 (5.6 and no EC).
Of course the above example is valid for lightening an image - the EC settings would be +0.3, +0.7, and +1.0. The lighter picture would be taken with an aperture wider than 4 - maybe 2.8.
You shouldn't think that the Exposure Compensation feature is redundant. Without it the fine exposure variations are impossible. Using the EC dial is quick and easy - basically you are able to make variations of the original exposure with a turn of a dial. Some cameras even allow you to "bracket" several EC variations - you press the shutter release button once, but the camera takes several pictures with different setting of the EC.
More advanced cameras that have a RAW mode allow fine adjustments of the EC to be done with the photo editing software. JPEG images (*.jpg) which are standard on most cameras cannot be digitally processed as much as RAW files and the wisest alternative is to take the best possible picture.
--Nikolay
Exposure Compensation 101 
Hybrid Mode
