HDR, or high dynamic range is a technique to capture more detail than your sensor can in a single shot. Basically the dynamic range of your sensor is the largest possible signal a pixel site can capture (without being pure white) or the lowest possible signal a pixel site can capture (without being pure black) with a single shot. Thus the limiting factor is the dynamic range of your sensor, within a given single photo.
High dynamic range is the name given to processing technique that results on a dynamic range larger than that which your sensor can capture in a single shot. The basic premise is that you shoot under-exposed for one shot, good exposure for a second one, and over-exposed for a third, then you combine those to create a photo that includes the furthest points of dynamic range from all three original photos, such that the resultant photo has more dynamic range than your sensor is capable of capturing in a single photo.
Now you can process one single capture into 3 exposures by using your PP software to under-expose and save, over-expose and save, and use those three photos to create an HDR, but often this is referred to as Psuedo-HDR.
some facts. The Nikon D800 has a dynamic range of 14.4 stops. Printed photos are about 6.5 stops and the human eye is somewhere between 20 and 25 stops.
By using three separate photos, you are using the entire available dynamic range of your sensor, to capture each photo.
Say your camera sensor has a dynamic range of 12 stops. But the scene you are taking a photo of has a dynamic range of 20 stops. You cannot capture all of that at once. So you capture 20-8 in your first, 16-4 in your second and 12 to 0 in your third, and using HDR processing create a photo that covers the whole twenty stop range. If you use one photo taken at say 16-4 and process that to create a psuedo-HDR, you are really only creating a photo with 12 stops, cause that is all you had to start with.