High Dynamic Range Photography (HDR) is just the process of taking multiple photos at different exposures and then combining them into one. The idea is that sometimes the Dynamic Range of a scene is too wide to capture in one image, as such you need to combine multiple images.
In plain english the dynamic range of a scene would be the light from the darkest dark area to the lightest light area. Just imagine you had a photo with a very bright part and a very dark part in the same scene. You would not be able to take just one image of that scene and have all the details show up in the darker are and the lighter area. This is what HDR is good for. Of course like all tooks HDR can be abused. Please HDR responsibly and don''t make horribly unrealistic and oversaturated images. Trust me, I've done that and they look awful!
Here are some examples of how I have done HDRs wrong:
(note that this photo didn't even need to actually be an HDR)
Here is another example of an HDR I made that is overdone but not has bad:
(note that this photo didn't even need to actually be an HDR either although it does make it seem overly dramatic)
Here is a video about making an HDR when shooting into the sun
Video by cmoeu
This is the software you need: Adobe Lightroom.