HDR stands for "high dynamic range". Camera sensors can only capture a limited number of greys between shades near black and shades near white. The difference between the darkest shade and lightest shade the camera can capture is the camera's "dynamic range".
The more "dynamic range" the camera's sensor can capture, the more you'll be able to see details in the shadows and highlights of your image.
Your eye can see a very high dynamic range. You can see into dark shadows inside a room while also seeing details in sunlit grass outside. A typical camera cannot see both at once. It can expose for the bright outdoors, or the dark indoors, but not both. So, to capture a "high dynamic range" image with dark shadows and bright highlights, you can take several photos, exposing one for the shadows, one for mid-tones, and one for highlights, then assemble them into a wide or "high dynamic range" original.
Unfortunately, you can't see that full range on today's monitors or save that full range in a JPEG image. This is why some cameras offer a "raw" image format with as much as 64 times more levels of brightness: so they can record much more of the original scene for you to work with later.
Web images are typically shared the JPEG format, has a limited number of steps between black and white. There is no such thing as an "HDR" JPEG image. There are only "tone mapped" JPEGs, where a high dynamic range scene or raw image had its colors and shades selectively mapped onto the limited colors and brightness range of a JPEG.
As a simplified comparison, a raw image can store 16 thousand levels of brightness per color, but a jpeg can only store 256 levels of brightness per color.
If you use a high end camera, it can capture a wider range of shades from black to white than a JPEG can show. Photo tools like "Adobe Camera Raw" try to convert the full range of shades in the raw image shot by the camera into a range that can be shown in a JPEG. The wide range of shades are compressed into a single limited range of shades. To keep the picture from looking flat and all grey, this compression is done using "tone mapping", where bright areas get contrasts added back in among nearby whites, and dark areas get contrast added back in among nearby blacks.
The same principle applies with an HDR. Between the dark, medium, and light exposures, there is too much brightness range for a single image, so the shades are compressed.
HDR software like Photomatix or Photoshop help you choose the tones you want to preserve, and choose how you want the contrast applied, to get an image that includes some lights and darks at the same time, while throwing away other colors and shades, to make the most of the limited dynamic range in a JPEG.
This brings us to the iPhone 4. In a very high contrast scene, some parts of the scene would be over-exposed or under-exposed, because the range between dark and light is too great.
I'll illustrate this with a picture taken to show a dynamic range problem and how to solve it.
In this picture, the patio stones to above left of the pool are in bright sun, and have become "washed out". The same is true of the fourth pillar or post. You can see wood tones on the third pillar, but the fourth one is washed out. Those brightnesses were too bright, so they just got mapped onto the brightness of white.
I've included this picture's "histogram" in the lower right corner. A histogram is a graph of how much of the image is each color from dark shades at the left of the graph to light shades at the right of the graph. In this histogram you can see there are a lot of dark blue tinted shades (all the stones in the shadows), and most of the image (reds, greens, blues) are below middle brightness. You can also see at the far right of the histogram the spikes in bright colors, with the highest spikes in the green (all the washed out bright green leaves).
iPhone 4 saves a regular exposure, then saves the composite image it makes from its dark and light exposure, which people are calling "HDR". This is the "HDR" photo the iPhone 4 generated from the same scene:
This image looks flat and grey. This is why some people are saying iPhone HDR images are dull, or washed out.
If you look at the histogram, you can see there are still some very dark blues (the stones and nearest post), but most colors have moved more towards the middle. In this case, the shadows of red, green, and blue, are now nearer the middle tones. We perceive middle tones as grey or dull. If you look at the right side of the histogram, you can see there is no longer a giant spike of bright white, the spike has been moved a little towards the mid-tones as well. The colors or brightnesses have been compressed towards the center, to be able to record a few more levels of shadow and a few more levels of highlight into the same limited 256 levels of brightness per color in the JPEG image format.
Studying the picture, you can see the areas that were washed out now have detail, such as the patio stones to the left of the bench, and the wood grain of the fourth post.
If you stop here, then yes, the iPhone 4 HDR photos are flat and dull. But you should not stop here. Your image has more of the original scene recorded in it, with more details in the shadows, and more details in the highlights.
Unfortunately to fit in the shadows and highlights, much of the colors were moved toward the flat dull middle of our perception. So, your job as a photographer is to "develop" the picture and decide which brightnesses and details are important to you, by adding back contrast. This iPhone HDR original has shadow detail you can push darker, and highlight detail you can push lighter. By adding back some contrast, you end up with a picture like this:
If you look very closely at the histogram, you can see some dark colors moved left into the shadows, some highlights pushed right into white, while keeping enough shadow and highlight detail to be able to see the stone pavings, grass and patio stones left of the bench, the green leaves of the pear tree, and the color of the fourth post in the sunlight. The overall image looks natural again, and is definitely better than the original non-HDR photo, even though they're both the same exposure or brightness.
iPhone 4 "HDR" mode preserves more of the shadow and highlight detail at the expense of a necessarily duller image, as a wider range than a JPEG can record is stored into the JPEG. You can then adjust exposure, brightness, and contrast of that JPEG to "expose" the dynamic range that's most appealing to you. You cannot fix the original exposure, because the highlights and shadows were cut off and lost.
You can adjust the iPhone 4 HDR image's contrast or tone mapping, and you absolutely should. Free software like iPhoto and Picasa can adjust brightness and contrast, as of course can Aperture or Lightroom.
Note: If you want the popular "HDR look" of wild contrast and halo effects, grab the ProHDR app for iPhone and go nuts. Those images that look like that are not actually "HDR" images, they are tone-mapped images. You can use the Pro HDR app to generate wild tone mapping, or to take sane exposures of complicated situations and blend or tone-map them in camera to your liking. The app works well as long as nothing in your image moves between shots.