One concept which is absolutely crucial to the understanding of exposure is that of the "mid-toned" object. This is an object of "middle" brightness, neither dark nor light. In terms of black and white, it would be a gray object, and more specifically an 18% reflective gray. This means that such a gray object would reflect 18% of the light which falls on it. Mid toned objects do not need to be such a "middle grey". They can be middle-green or middle-brown or any other mid tone color. An "average" natural scene, consisting of grass, trees, rocks etc. is in fact mid toned. That is to say, the exposure required to render an 18% gray card in the scene as a mid tone gray in the image would also render the rest of the scene correctly.
Though the human eye can see tonal detail in objects over a wide brightness range (at least 1:1000 or about 10 stops), photographic recording media (film or digital sensors) have a more limited range. Most 35mm (and APS-C) digital DSLRs have a usable dynamic range of around 8-9 stops. This is more than slide film had (about 5-6 stops), but less than the capability of negative film, which can in some circumstances exceed 15 stops.
With a 5 stop range (which may be the case for high contrast slide film) it can only record detail over about a 1:32 range of brightness. If we chose an exposure for a scene which reproduces a mid toned subject as a mid toned image on the film, any object which is 2.5 or more stops brighter will be reproduced as a featureless "washed out" very bright tone and any object which is 2.5 or more stops darker will be reproduced as featureless "blocked up" very dark tone. With current DSLRs, you can get shadow detail in areas which are 4 stops darker than a mid tone, and highlight detail in areas which are about 4 stops brighter (especially if you work with RAW files and/or 16-bit images).
So just how do we determine the "correct" exposure for even a mid toned object, let alone a very bright or very dark one? Well, we use an exposure meter of some kind. There are two types of exposure meter, incident light meters and reflected light meters. The incident light meter measures the amount of light which falls on a object to be photographed. Most hand held exposure meters are incident meters (or have an incident mode) The meter is placed next to the object, in the same light as the object, and pointed at the camera position. The electronics in the meter then calculate what the correct exposure would be to render a mid toned object as a mid toned image. With this calculated exposure, light toned objects will form light toned images and dark toned objects will form dark toned images. A reflected light meter on the other hand measures the amount of light reflected from an object. All light meters built into cameras are reflected light meters. The meter electronics calculates what exposure would be needed to render this amount of reflected light as a mid tone. One obvious problem here is that the object may not be a mid toned object! It could be a white polar bear or a black cat. This is one problem with reflected light meters, and this is the type of meter which all SLR metering systems are based on.
Even with incident meters there can still be problems, especially when the recording medium has a limited dynamic range. As mentioned above, the total tonal range of some slide film is only about 5 stops. If an 18% reflective gray card is mid tone, then a 100% reflective white object will be about 2.5 stops brighter. If exposed for a mid toned object, a white object will appear on the film as a featureless "washed out image". In order to record a white object on film and still be able to see some texture in the image (e.g. the plumage of a white bird), we need to reduce exposure slightly, typically by one stop. Similarly, for a very dark object, if we use the incident light meter reading, the dark object may fall outside the range in which slide film can record detail. If we want to record detail in, for example, the fur of a black cat, we may need to open up one stop or so from the exposure indicated by the incident meter.
Under clear sunny skies, when the sun is reasonably high in the sky, an incident light meter will always give the same reading. That reading is the basis for the "sunny f16"rule which states that the correct exposure for a front light (not side light or back light) mid-toned object is 1/ISO at f16 where ISO is the ISO setting. Thus at ISO 64, the correct exposure is 1/60 at f16 (or 1/125 at f11 and so on). Stopping down a stop for very light subjects and opening up a stop for very dark subjects is a rule which still applies. An incident reading is an incident reading, whether you get it from in incident light meter or the sunny-f16 rule. If you are using the sunny-f16 rule (which applies for front light subjects) and need to photograph a side light subject, open up one stop. For a backlit subject, open up two stops. Thus for a backlit white subject you would close down a stop because the subject is white and open up two stops because it is backlit, giving a final exposure of one stop open from sunny f16. Note that if you are using a real incident light meter for side and backlit subjects, you don't need to add an exposure correction for the lighting direction - the direction in which you point the meter (towards the camera position, which for side and back lighting means away from the sun) will result in correct exposure readings.
Most modern cameras (and all DSLRs) have a third metering mode, known by various names such as "matrix" or "evaluative" metering. In this mode the image is split up into many different "zones", from as few as 3 zones to 50 or more. The camera analyzes the brightness of each zone and uses an internal computer to decide what the most probable "correct" exposure is. Photographers should be very careful in using exposure compensation with these modes, since there is no why to know whether or not the camera's computer has already tried to do this for itself! For example, take a white bird sitting in full sunlight. Since most metering modes are set up for mid toned objects, some meter correction would probably be needed for this subject. The thinking photographer might open up one stop from a reflected light reading so the bird recorded as white - not middle gray, or stop down one stop from an incident light reading - so the feather detail in the pure white bird would not be washed out. Some matrix metering systems might recognize the fact that the brightness of the light reflected from the bird was so high that it could only come from a white subject in full sunlight and therefore it would open up a stop by itself! You just can't predict what an evaluative metering system is doing, so you basically have to trust it if you use it.
The exposure histogram is just a pictorial representation of how many pixels in the image have a particular luminosity level. Lets say for simplicity's sake that the pixels can have luminosity levels from 0 (black) on the left to 255 (white) on the right. The vertical height of the plot is proportional to the number of pixels with the given luminosity.
The histogram on the left shows a pretty well exposed image. The vertical height of the plot shows the number of pixels and the horizontal scale shows their luminosity from black at the left to white at the right. The goal of exposure is to try to get the brightest pixels close to the right side of the histogram. If you do this (without any pixels at the far right end, corresponding to the maximum luminosity level of 255), you will have detail recorded even in the brightest parts of the image. If you overexposure you may get a histogram which looks something like that on the right. Her you can see a significant number of pixels at the highest luminosity (white) end of the range, indicating that detail has been lost in the image highlights. One caution when using histograms to judge exposure is that if you only have a small highlight area in the image, you won't have many pixels there. The histogram isn't very sensitive to small areas and you may not be able to tell if you've blown those highlights out. So the rule is that you don't want a big empty space on the right, but you don't want to see an obvious spike at the far right edge, corresponding to saturated pixels with a luminosity value of 255.
Some cameras have the capability to show individual histograms for the three color channels (Red, green and blue). This gives more insight into exposure and by examining them you can avoid "blowing out" the image in one color (i.e. pushing the brightest pixels all the way to the right edge of the histogram), which sometimes doesn't show up in the overall luminosity histogram if the other two channels have lower luminosities. This can happen when photographing very brightly colored subjects. For example when shooting a bright red rose, it's fairly easy to lose color detail in the flower by blowing out the red channel. This may be fairly easy to see in the red channel histogram, but may not be at all obvious in an overall luminosity histogram which combines all three color channels.
Probably the best way to learn about exposure in the digital age is to look at the EXIF data which just about every digital camera records as part of the image. EXIF stands for "Exchangeable image file format" and this data can be read by just about all image editors and digital photo organizer programs. Even the most basic cameras should record things like the shutter speed, aperture, ISO setting, exposure compensation etc.
Once you have conquered the control of exposure, you'll have one less aspect of photography to worry about and all you have to do is get the composition right and find your subjects and you'll have those perfect pictures we all dream of. If only that were that simple...