I realize that an image histogram is a graphical display of an images tonal distribution (i.e. horizontal darks to lights, vertical pixel distribution), but how does one really use it and why? I mean, can't you determine everything you need just by looking at the image?
While there may not be a "right" answer to this question, there are "correct" answers. A histogram is a powerful tool, and when you understand how to use it effectively, it can greatly help your photography.
As you mentioned, a histogram is a representation of tonal range and distribution in a photo. The basic mechanics are as such:
(As a real (float) number, the values of a 32 bpp HDR image range from 1.0 x 10^-37 through 1.0 x 10^38. In more real-world numbers, tonal range from black, through very dim starlight (0.00001), through indoor lighting (1-10), through the sunlit outdoors (1,000,000), to the brightness of the sun itself (100,000,000) and well beyond. All those values can be represented in a single HDR image.)
Given these facts about a histogram, there is a wide variety of information you can gleen from one:
Contrast is the measure of difference between the brightest tones and the darkest tones. The more range a histogram covers between its left and right edges, the greater the contrast of an image:
Key & Exposure
Key is the rough measure of brightness in an image, with high-key being brighter, and low-key being darker.
(A histogram riding up the right-hand side of the histogram probably indicates overexposure – clipped highlights. A histogram riding up the left-hand side of the histogram probably indicates underexposure – blocked shadows.)
When using a colored histogram, the convergence of red, green, and blue peaks is an indication of white balance. In particular, the offset of major blue peaks can be a strong indicator of the warmth or coolness of a photo:
In a properly white balanced image, blue is usually a little right of red and yellow peaks.
The balance and height of peaks in a histogram is an indication of tonal range and tonal balance. Parts of the histogram that are very low (valleys) indicate very low volume for those tones. Parts of the histogram that are very high (peaks) indicate very high volume for those tones.
A basic colored histogram will often show gray, red, blue, and green. A more advanced colored histogram may also show yellow, magenta, cyan.
Colored peaks are an indication of the volume of those given primary colors, the horizontal position of a colored peak is an indication of the tone of colors of that particular primary or primaries.
Gray indicates a balance of primary colors at those tones. Off-primary color peaks (or partial height lines), such as yellow, magenta, and cyan, indicate a blend of two primary colors at those tones.
As mentioned by Jordan H., there is a trick called "expose to the right" (or ETTR) that can be useful to get you the optimal RAW data. When shooting a scene, particularly those that have a broad range of contrast that may be on the border of, or possibly slightly beyond, the 5-6 stop dynamic range of a digital camera, capturing enough tonal range in the shadows can be difficult.
This is due to the the limitations of most current digital sensors, and how they are more sensitive to highlights than shadows. "Exposing to the Right", which is a technique where you slightly overexpose your shots by 1/3 to 1/2 of a stop (which, in turn, shifts your histogram to the right...toward highlights), can help mitigate these limitations.
Exposing to the right can also help alleviate noise problems in the shadier parts of your images. It should be noted that exposing to the right requires that you use RAW format, as only with raw are you saving enough information to correct your overexposure during post-processing to bring your image back into normal range. The benefit of this technique is that it allows you to capture detail that would otherwise be lost, without the need to resort to ND grad filters or other more extreme measures.
This guideline is just that, a guideline. With newer camera sensors, dynamic range is improving, and capturing a greater range of contrast in a scene with a single shot is easier. However, even as digital sensor dynamic range improves, there will always be times when we need to shoot "on the edge" or what is possible, and tricks like shooting to the right will always be useful.
In the old fashioned darkroom we had a tool called the densitometer. It measured the density (how much light is blocked when you shine a light through) of the negative or slide. It was a bulky and expensive device, and of course it required the film to be developed so it wasn't very practical for use in the field. But we could use it in testing to determine optimum exposure and development techniques for personal use.
Film photographers are primarily concerned with a lack of density. Too much density can conceivably be burned through (apply a greater intensity and/or duration of light to a local area) to recover detail in printing. A lack of density (shadows with negs, highlights with slides) cannot be corrected for, as it means the detail/information isn't there to recover.
Digital photographers have to be concerned at both ends. When the tonal range goes beyond the edges of the histogram scale it means there is nothing in those areas except white or black. You can lighten or darken, but you are only making solid gray tones with no detail recovery possible. I use the histogram display like a field densitometer. It shows me if I've got areas of my photo where details will not be recoverable no matter what Photoshop tricks I know. Because I use ETTR (expose to the right) I don't expect my files to come out of the camera looking like the finished photo any more than I expected my negs to look like my finished photos. The histogram helps me assess the exposure even if what's on the LCD doesn't look quite right.
As was mentioned in the comments, the histogram display is created from a jpeg processed according to the in-camera software parameters even if you are shooting raw. Raw is like exposed but undeveloped film. There is a potential image there, but we can't see it until it is processed. The raw data needs to be processed for a histogram to be generated. I keep my cameras set to the most neutral in-camera processing parameters available in order to get a more accurate histogram. It's still not the same as the histogram I'll see in Adobe Camera Raw when I open the files there. The standard default in-camera processing, among other "Picture Styles" or whatever your brand calls them, is often higher contrast and higher saturation. Both of these can cause a histogram from an in-camera jpeg to display the histogram falling off the highlights side (right side), when that is not the case with the actual raw data, and the potential image if processed with other software or techniques. Only practice and experience will teach you how to interpret the histogram for your processing techniques, and how much of the blinkies you can ignore.
To add to jrista's excellent answer, an histogram is god-send in bright situations where your LCD is very hard to read (eg: at noon in a field of snow, for example).
I'm not a technical photographer and never made much use of the histograms that I've been seeing for the last 5 or 6 years, until this summer. I went to Israel with family this summer, and as the goal of the group was to see a lot of sites, not to be in various places with ideal photographic lighting, I took a lot of photos in midday sun. To make matters harder, many of the historic sites in Israel consist of ruins built of light colored stone. Light colored stone, midday sun: you get the picture (or not, as the case may be).
It took a lot of post-processing to get the pictures to look like anything I was willing to show anyone, and as I played with the various contrast and light settings in Picassa, I noticed that by the time I got the photo to where I wanted it, I had made certain changes to the histogram: I had spread it out from being bunched up to the right to spread out that curve over the full range, and then the scenery looked pretty good. Alternatively, when I was shooting people in the context of the white rocks, the way to get those shots to come out was to take the bump on the left side and spread that out over the entire area (over-exposing the background, but getting good contrast on the faces). Over the course of processing about 1000 photos, I began to understand the histogram and how to use it as a tool, even in less extreme situations.
I know I'm not as technically proficient as many of you, but this is how I, as an amateur, came to grips with the histogram.
The basic thing I use it for is telling at a quick glance if there is enough light. If the histogram is all on the left, it is too dark and at best you will need to do some post-processing tweaks to get a usable photo. If it is all to the right, it will probably be washed out.
Note that you can recover somewhat from both of these in post-processing, but you will almost certainly have lost some detail.
When you look at a picture, it can be hard to tell exactly how well the shadows and highlights fill up the range, especially if you are looking at the picture in the display of the camera. The histogram gives you an exact measure for how the image is exposed.