"HDR stands for High Dynamic Range imaging, and it's an old photography practice recently introduced to cameraphones like the iPhone and some Android devices (or with the use of special apps). You're on the right track: it's supposed to make your pictures look better, but it depends on when you use it. Here's a quick primer on how HDR works, and when you should—and shouldn't—turn it on."
HDR means High Dynamic Range, and when it is active on an iPhone and you take a photo, it actually takes three photos very quickly with different exposures. That's why it takes slightly longer to take a picture when HDR is on. After you take it, the iPhone's photo software combines the three images into what theoretically is an image with better contrast in both light and dark areas of the image.
It doesn't work very well with moving images, but it can be very helpful with backlit subjects and landscape photos.