HDRI stands for “High Dynamic Range Imaging”, and it basiclly means that an HDRI image can represent light brightnesses that are brighter than the brightest value your monitor can display. It also means that they just generally have higher accuracy, since they aren’t limited to the 256 values per channel that most image formats are limited to.
HDRI images are useful for lighting 3D scenes (usually as a sky texture) because they are capable of representing a much, much, much larger range of brightnesses (with higher accuracy) than normal images can.
Let me give an example. Let’s say you have picture of a bright sunny day that you want to use to light your scene with. There is no way that you can accurately represent the brightness of the sun in the image, because it is so much brighter than everything else in the picture. A normal image will typically just represent such brightness as the brightest white the monitor can display. But that’s still no where near the brightness of the sun. If you were to light your scene with that image, you would certainly end up with lighting that’s too dim and diffuse.
If, however, you use an HDRI image, the brightness of the sun would be accurately represented as many many times the brightness of the brightest white the monitor can display, and thus the lighting in of the 3d scene would be more accurate.
Obviously, HDRI images aren’t useful for pictures you just want to look at (the monitor isn’t capable of displaying them correctly anyway), but they are very useful for image-based lighting.
Hope this helped!