HDR stands for High Dynamic Range. The “dynamic range” is qualifying the variation between the lightest highlights and darkest shadows.
Typically, when we focus on these contrasting opposite areas, in order to change the amount of perceived light, a camera sensor would change its ISO setting, a lens would change its aperture or similarly, even our eyes pupil would dilate. That is to say, we cannot expect to see or capture bright areas (like the sky) and darker areas (such a porch under a roof) with the same amount of light.
Bracketing
The standard and most qualitative way of obtaining a HDR rendering is achieved by taking multiple photos (often 3) at different exposure, one below and one above your target exposure, and then to combine all three with a dedicated software.
Due to this, you will need a tripod in order to avoid the blur induced by your hand movements. Obviously it works best with a static subject such as a landscape, but you can still achieve a similar effect by using shadow and highlight tool.
Now most smartphone photo applications have a HDR setting which can also be used to take several exposures in order to achieve the same bracketing effect. Some phones are able to apply a HDR kind of style with only one exposure, by applying an algorithm to boost shadows.
Have you noticed that special appeal that iPhone photos have in terms of lighting and colors ? Now you know why: iPhone are post-processing their sensor input to balance light and shadows in a similar fashion as the shadow and highlight tool.
When to use: this technique works best with landscapes, especially when the sun is low or during nighttime, or with indoor situations. Bear in mind the HDR kind of style will not be realistic and when pronounced, it will create whitish backgrounds around bright / contrasting subjects. You can temper this side effect by lowering the difference of exposure between shots to combine.
Many photographers also use HDR to highlight metallic surfaces such as on cars and garage tools.