Camera sensors on phones have come along way in just the last handful of years. Once upon a time it was a marvel to have any sort of camera on your phone at all, much less one that could even be considered remotely good quality. Nowadays we nitpick every little thing about the image quality on the cameras found on our smartphones, down to the nitty gritty details in the image. 13 megapixel sensors have come and gone as the hot new thing and this year it looks like we're likely to have 16 and 20 megapixel sensors as standard on every flagship phone you can think of. This push for higher and higher image quality usually comes at a cost in that the pixels themselves get smaller, allowing for less light to come in to each one at a time.
This creates a problem with dynamic range, or the measurement between the brightest and darkest pixels viewable in the scene. Often times on smartphones you'll find that looking at a bright part of the scene makes the entire rest of the scene completely dark, or the opposite happens when focusing on a dark part of the scene and all the light parts of the scene become blown out. This is where high dynamic range, or HDR, comes in. Traditionally HDR works via a method called exposure bracketing, which means that the camera takes a picture at high exposure, grabbing all the detail in the darker parts of the scene, and one at low exposure, grabbing the detail from the brightest parts of the scene, and stitches the two together. HDR is essential for getting the absolute best picture possible out of your smartphone, and it's really the only way to negate physics and the constant shrinking of pixels that's become necessary to pull more detail out of a scene.
More than likely you've used HDR before but maybe didn't know exactly what to use it for or how to use it. Depending on your phone HDR is only useful in a few situations, particularly ones with no movement at all. Since the phone generally has to take two pictures right after another, if there's any movement you'll get a ghost image of the moving object created from the two images being stitched together. Google's HDR+ method, found on the Nexus 5 and 6, helps alleviate this problem by taking samples of light sources instead of using exposure bracketing. While this doesn't completely fix the issue of ghosting in scenes with movement it significantly helps, but what about those without HDR+? Let's go over a few simple steps to ensure that HDR is working best for you in as many situations as possible.
First off HDR should usually be used in bright areas only. This is because the shutter speed has to slow down, allowing more light in but exacerbating the natural shake of your hands. This is doubly true in HDR when it's got to take more than one image and stitch them together, making an image that looks just plain blurry. While you can attempt to do this with phones that feature Optical Image Stabilization, or OIS like the LG G2, G3, Samsung Galaxy Note 4 or Nexus 5 and 6 have, it's definitely something that's not going to work out as well on phones that don't.
HDR enhances the image best in areas of extreme light and dark, so taking a picture of sun shining on an object would pull out the detail in both the bright areas and the shadows. HDR will still give you better results in evenly lit areas but it won't be as dramatic of a difference as if you were in an extreme contrast situation. Some phones will even give you the option to save both the original and HDR shot so you can compare afterward and see the difference.
Lastly make sure you hold as still as possible when taking an HDR shot. Some phones are faster at taking the multiple shots than others and will be more forgiving, but in general resting your arm on something is going to be your best bet. Obviously being in a moving car or some other situation where you'll run into shake no matter what you do probably isn't a candidate for an HDR shot. This includes moving objects which will become blurry if moving too quickly. Slow moving objects may be just fine, so take a shot and give it a try! You may also find scenarios where keeping HDR off is preferable simply because the image with higher contrast looks better, and that's all part of what makes photography so fun and challenging. The beauty of digital cameras is that you don't have to worry about wasting film, so take those shots and quit worrying!