HDR (high dynamic range) is basically when you shoot the same scene with 3 different shots; 1 at normal exposure, one at a lower exposure, and the third at a higher exposure. You would then combine them to get a picture with the highlights NOT blown out to white (think of the sky) and shadows dark but able to see fine detail. This is in terms of photography, not video.
In terms of video, and what is seen in new 4k HDR HDTVs, the major benefit is having less banding. HDR video is encoded with HEVC (or x265) on new 4k blu-rays at 10bit, instead of the normal 8bit.
You can encode traditional 8bit video (current regular blu-rays) at x265 10bit and get very good compression while maintaining very high quality. I’ve been encoding some of my own personal blu-rays to x265 10bit and the results are quite good.
http://x264.nl/x264/10bit_02-ateme-why_does_10bit_save_bandwidth.pdf (PDF file)
Also, here is a short x265 10bit clip I did of the DeSpec 2.7. The original was around 185MB, this is only 35MB. VLC and MPC-HC can play this file.
https://drive.google.com/file/d/0B9tjpukbbRggZWdtdlhNNzduT0k/view?usp=sharing (MKV file @ 720p)
x265 10bit still has some trouble with maintaining grain, but there wasn’t a whole lot in the original v2.7.
As for encoders, I use Staxrip.
If someone else can offer more insight (and correct me if I’m wrong), that’d be great.
Hope this helps!