An HDRI is a High Dynamic Range Image,
It's basically a digital image format designed specifically to be able to allow for more accurate storage of brightness values. For example:
A normal 24-bit image only gets you 256 levels of brightness, 0-0-0 RGB value being black, and 255-255-255 RGB value being white. This is sufficient to show a nice picture on your computer monintor because the monitor doesnt show more then 256 levels of brightnes anyways, but not sufficient for actual accurate storage of brightness values.
For example:
In a 24bit photo, a lightbulb and the sun could both come out white (255-255-255 RBG value). Yet, in reality, the light comming from the sun is much more intense (thus brighter) then the light comming from a lightbulb. The 24bit image is just not capable of expressing this level of contrast in the amount of brightness, it clips at 255-255-255 RGB.
To accurately store the difference in brightness of the lightbulb and the sun, one would use an HDRI image because this type of image is designed to allow for much, much more brightness values. (think towards thousands and thousands of brightness values here, maybe tens of thousands, i'm not entirely sure how much exactly)

So why would one use HDRI's if a normal monitor isnt capable of showing all the brightness values in the image? well here's the explaination for that:

In 3D renders using IBL (image based lighting), an image "wrapped around your 3d scene" can be used to simulate light cast by lightsources, or bounced off objects, surrounding your 3D scene. For instance the sky, clouds, sun, buildings in the street, etc.

To accurately simulate this light, it suddenly becomes neccesary that the image you use is in fact able to express the different degrees of for instance the brightness of a lightbulb and the sun which i gave as an example because these really do affect the way your scene gets lit. If you turn on a light in a dark room it lights the room, but never as brightly and vibrantly as the sun would, shadows stay darker, as do various more or less obscured area's, nooks and crannies. It is in effects like this that the extra brightness values in an HDRI image turn out to be extremely usefull to accurately portray/simulate the real life situation.

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