This first image (of the North End in Boston) was taken with the normal iPhone camera application.
As you can see it does improve the image, brightening up the dark (underexposed) areas, while equalizing the bright (overexposed) areas so that you get the detail you are looking for. The HDR process takes two images, so you may see things like the same car in two different places. Something to be aware of.
This is a very basic example of HDR, and the image was created with software that runs on the iPhone. This software only takes two images, one underexposed and one overexposed. Most HDR pro's take at least 3 images. You can do some pretty amazing things with a real camera and some real image editing software or HDR software. For some amazing images, take a look at this Google Search for HDR Images.