Stands for High Dynamic Range Imaging.
This is a method of creating images that employs 32bit floating point colour to display a far more dynamic range of colours than a computer moniter is capable of displaying. There are several image formats that are capable of using this technololgy (.tiff is one that already existed, .hdr was developed specifically for HDRI). The advantages of this is that it allows light sources to be accurately represented digitally.
Allow me to explain: If you take a picture of you sitting in front of your computer reading nodes and scan it into a computer, the range of brightness and colours is 8 bits per channel. That means the brightest things in the picture will have their luminosity capped at 256 (or 255 as the case may be). This isn't very realistic, as the computer your staring at may be bright in the range of colours in your photo, but is about one billionth the brightness of the sun. If you took a picture of the sun, it would also have its brightness capped at 255 in traditional 24bit imaging. Thats where HDRI comes in. HDRI can take a high dynamic range of colours. This is incredibly important in Computer Animation because an HDRI image can be used to light an entire scene. If the HDRI image is used as a background, then using Radiosity Rendering, the power of the lights in the HDRI image can be accurately represented and diffused on the surface of the object to be rendered, creating a new level of photorealism. It also assures perfect compositing, every time.